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SLAUGHT  AND  LENNBS 


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http://www.archive.org/details/elementaryalgebrOOslaurich 


Roger  Bacon 

(1214-1294) 

"  All  sciences  rest  ultimately  on  mathematics/' 

"  Mathematics    should    be    regarded    as    the    alphabet    of    all 
philosophy."' 

"  Divine    mathematics,    which    alone   can   purge   the   intellect 
and  fit  the  student  for  the  acquirement  of  all  knowledge.'' 


ELEMENTARY    ALGEBRA 


BY 


H.    E.    SLAUGHT,    Ph.D.,  ScD. 

PROFESSOR   OF   MATHEMATICS    IN    THE    UNIVERSITY 
OF   CHICAGO 

AND 

N.   J.    LENNES,    Ph.D. 

PROFESSOR   OF   MATHEMATICS    IN   THE    UNIVERSITY 
OF   MONTANA 


•  ,  •  •     •• 

•  •   .  .  •  , 


3i«<c 


ALLYN   AND   BACON 

Boston  Neto  gorit  (frt}irag0 


G  Ai^4- 


COPYRIGHT.  1915, 
BY  H.  E.  SLAUGHT 
AND  N.  J.  LENNES 


AAN 


Nortoooti  PrfBB 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Oo. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

The  Elementary  Algebra  is  planned  to  cover  the  work  of  the 
first  year  in  this  subject.  It  is  in  no  sense  a  revision  of  the 
authors'  First  Principles  of  Algebra,  but  a  new  book,  designed 
to  meet  the  most  exacting  requirements  of  college  entrance  or 
other  examination  boards  and  the  syllabi  of  various  states. 
The  presentation  of  topics,  therefore,  follows  the  traditional 
order. 

The  new  Algebra  contains  numerous  attractive  features,  all 
aiming  to  make  the  subject  more  simple  and  interesting  and 
therefore  more  valuable  to  first  year  pupils.      Among  these 
features  the  most  distinctive  are  perhaps  the  following  four : 
The  presentation  of   the  subject   is  as   simple  as 
it  can  be  made. 

The  book  is  equipped  with  an  unusually  full  and 
complete  set  of  exercises  and  problems. 

Vital  purpose  is  given  to  the  study  of  algebra  by 
using  it  to  do  interesting  and  useful  things. 

Emphasis  is  given  to  the  human  interest  of  algebra. 

To  take  up  these  four  features  in  detail : 

The  simplicity  of  the  book  is  especially  shown  (a)  in  the 
careful  explanation  of  each  new  subject,  which  is  fully  dis- 
cussed, defined,  and  illustrated  ;  (h)  in  the  gradation  of  topics, 
—  so  carefully  managed  as  to  be  almost  imperceptible  to  the 
pupil ;  and  (c)  in  the  gradual  and  natural  introduction  of  the 
symbols  and  processes  of  algebra. 

Each  new  topic  is  fixed  by  a  large  number  of  simple  illus- 
trative examples  and  exercises,  oral  and  written,  and  finally 

iii 

460003 


iv      ,  PREFACE 

by  practical  problems.  These  exercises  are  unusually  numerous 
and  thorough,  and  there  is  at  the  end  of  the  book  a  complete 
set  of  reviews  for  every  chapter,  followed  by  twenty  pages  of 
miscellaneous  problems. 

The  interest  and  value  of  the  subject  are  emphasized  by 
showing  (a)  how  algebra  grows  out  of  arithmetic,  (6)  how 
much  easier  it  is  to  solve  problems  by  algebra  than  by  arith- 
metic, and  (c)  by  applying  these  algebraic  solutions  to  count- 
less problems  of  every-day  experience,  as  well  as  to  those 
taken  from  other  studies  in  the  school. 

To  emphasize  the  human  interest  of  algebra,  numerous  his- 
torical notes  have  been  introduced,  showing  how  many  scholars 
have  labored  through  the  centuries  to  develop  this  subject. 
Also  there  are  portraits  of  well-known  mathematicians,  with 
accounts  of  their  lives  and  of  their  contributions  to  mathe- 
matics, especially  to  algebra.  A  briefer  course  may  be  made, 
if  desired,  by  the  omission  of  certain  topics  which  are  marked 
in  the  text  with  a  star.  These  omissions  will  not  interrupt  the 
continuity  of  the  work. 

It  is  hoped  that  the  Elementary  Algebra  will  be  found  a 
practical  and  satisfactory  instrument  for  the  study  of  the  sub- 
ject, and  that  it  will  still  further  extend  the  coyicrete  study  of 
algebra,  with  which  the  authors  have  been  identified. 

H.  E.  S. 
N.  J.  L. 

June,  1915, 


TABLE   OF   CONTENTS 


I. 


IL 


Introduction  to  Algebra. 

Letters  used  to  Represent  Numbers   .... 

Algebraic  Expressions        ...... 

Addition  and  Subtraction  of  Numbers  having  a  Common 

Factor 

Multiplication  of  the  Sum  or  Difference  of  Two  Numbers 
Division  of  the  Sum  or  Difference  of  Two  Numbers  . 

Multiplication  of  a  Product 

Division  of  a  Product  ...         .         .•        . 

Order  of  Indicated  Operations     ..... 

Equations  and  Problems. 

Solution  of  Equations  ...... 


Directions  for  Written  Work 
Solution  of  Problems 


III.  Positive  and  Negative  Numbers. 

Addition  of  Signed  Numbers 
Averages  of  Signed  Numbers     . 
Subtraction  of  Signed  Numbers 
Multiplication  of  Signed  Numbers 
Division  of  Signed  Numbers 
Interpretation  and  Use  of  Negative  Numbers 

IV.  Addition   and   Subtraction   of  Algebraic    Expres 

signs. 

Addition  and  Subtraction  of  Polynomials   .         .  .  . 

Algebraic  Expressions  in  Parentheses         .         .  .  , 

Forming  Algebraic  Expressions  .         .         .  .  . 

Problems  on  the  Arrangement  and  Value  of  Digits 

V 


12 
14 
16 
IS 
20 


25 
30 
32 


45 

49 
50 
56 
59 
63 


69 

75 
79 


VI 


TABLE    OF   CONTENTS 


OUAPTEB 

V.     Multiplication  and  Division  of  Algebraic  Expres 

SIONS. 
Multiplication  of  Polynomials  .... 

Product  of  Two  Powers  of  the  Same  Base 
Quotient  of  Two  Powers  of  the  Same  Base 
Division  by  a  Polynomial        ..... 

VI.     Special  Products  and  Quotients. 

The  Square  of  a  Binomial        ..... 
Product  of  the  Sum  and  Difference  of  Two  Numbers 
Binomials  with  First  Terms  Alike    . 
The  Square  of  a  Trinomial      .... 
The  Cube  of  a  Binomial ..... 
Quotients  derived  from  Special  Products 
Dividing  the  Sum  or  Difference  of  Two  Cubes 
Using  Special  Products  in  Equations 

Vn.    Factoring. 

Case  I  :   Monomial  Factors 
Case  11  :  Trinomial  Squares    . 
Case  III  :  The  Difference  of  Two  Squares 
Case  IV  :  The  Sum  of  Two  Cubes  . 
Case  V  :  The  Difference  of  Two  Cubes    . 
Case  VI  :  Trinomials  of  the  Form  x^  -\-px  +  q 
Case  VII  :  Trinomials  of  the  Form  ax-  +  hx  -{-  c 
Case  VIII  :  Factors  found  by  Grouping    . 
Case  IX  :  The  Square  of  a  Trinomial 

*  Case  X  :  The  Remainder  Theorem  or  Factor  Theorem 

*  Factors  of  x"  +  2/"  and  x"  —  ?/'» 
VIII.     Equations  Solved  liY  Factoring 

Quadratic  Equations 
Equations  of  Higher  Degree    . 
Problems  solved  by  Factoring 

IX.     Common  Factors  and  Multiples 

Highest  Common  Factor 
Lowest  Common  Multiple 

X.     Algehraic  Fractions. 

Reduction  of  Fractions  to  Lowest  Terms 
Reduction  of  Fractions  to  a  Common  Denommator 
Addition  and  Subtraction  of  Fractions 
Multiplication  of  Fractions       ..... 


84 

92 

96 

102 

108 
110 
112 
114 
116 
117 
118 
119 

122 
124 
126 
128 
129 
130 
133 
136 
137 
138 
141 

147 
150 
152 

155 

157 

162 
164 
168 
172 


TABLE   OF   CONTENTS 


Vll 


Division  of  Fractions 
*  Complex  Fractions 


XI. 


XII. 


XIII. 


XIV. 


XV. 


XVI. 


Equations  involving  Fractions. 

Clearing  of  Fractions       ...... 

Problems  leading  to  Fractional  Equations 

Ratio  and  Proportion. 

Important  Properties  of  a  Proportion  —  Cases  I-VII 
Application  to  Similar  Triangles      .         .         .         . 


Literal  Equations  and  their  Uses. 

Advantage  in  using  Literal  Equations 
I.     Interest  Problems   . 
II.     Problems  involving  Motion     . 
HI.     Problems  involving  the  Lever 
IV.     Miscellaneous  Literal  Equations 


Simultaneous  Equations  of  the  First  Degree. 

Definitions       ...... 

Elimination  by  Addition  or  Subtraction    . 
Elimination  by  Substitution 
Simultaneous  Fractional  Equations 
Simultaneous  Literal  Equations 
Problems  leading  to  Fractional  Equations 
Problems  involving  Two  Unknowns 
Simultaneous  Equations  in  Three  Unknowns 
Problems  involving  Three  Unknowns 


*  Graphic  Representation. 

Graphs  of  Statistics  .... 

Graphic  Representation  of  Equations 

Square  Roots  and  Radicals. 

The  Square  Root  of  a  Monomial 

The  Square  Root  of  a  Polynomial  . 

The  Square  Root  of  an  Arithmetic  Number 

Simplifying  Square  Roots 

Equations  solved  by  Square  Roots 

Applications  of  Square  Root    . 

Problems  involving  Square  Roots  . 


PAGE 

175 
178 


182 

185 


189 
193 


196 
197 
199 
202 
204 


207 
208 
211 
213 
217 
218 
220 
224 
227 


230 
235 


240 
243 
247 
252 
256 
257 
258 


Vlll 


TABLE   OF   CONTENTS 


CHAPTEB 

XVII.     Further  Operations  on  Radicals. 


Definitions  ...... 

Reduction  of  Surds       .... 

Addition  and  Subtraction  of  Surds 
Multiplication  of  Surds 

*  Rationalizing  Binomial  Denominators 
Equations  involving  Radicals 

XVIII.     Quadratic  Equations. 

Completing  the  Square 

Checking  Results  in  Quadratic  Equations 

Solution  of  the  Quadratic  by  Formula . 

*  Imaginary  Numbers  .... 

*  Quadratics  with  Imaginary  Roots 
Fractional  Equations  leading  to  Quadratics 
Equations  in  the  Quadratic  Form 
Equations  containing  Radicals 
Problems  involving  Quadratics     . 

XIX.     Systems  of  Quadratics. 

One  Quadratic  and  One  Linear  Equation 

*  Systems  of  Homogeneous  Quadratics 
Problems  involving  Systems  of  Quadratics 

XX.    *  The  Binomial  Formula. 

Mathematical  Induction 

The  General  Term        .... 


XXI. 


*  Variables  and  Functions. 

Direct  Variation   . 
Inverse  Variation 


XXII.     Review  Exp:rcise8. 

Review  Exercises  on  Chapter  I    . 
Review  Exercises  on  Chapter  II  . 
Review  Exercises  on  Chapter  III 
Review  Exercises  on  Chapter  IV 
Review  Exercises  on  Chapter  V  . 
Review  Exercises  on  Chapter  VI 
Review  Exercises  on  Chapter  VII 
Review  Exercises  on  Chapter  VIII 
Review  Exercises  on  Chapter  IX 


262 
263 

268 
270 
274 
276 


278 
281 
283 
284 
287 
287 
289 
290 
291 

297 

301 
305 

310 
311 

314 
316 


318 
319 
320 
321 
322 
323 
324 
325 
326 


TABLE   OF   CONTENTS 


IX 


CHAPTER 

XXIL 


Review  Exercises  {continued). 

Review  Exercises  on  Chapter  X  . 
Review  Exercises  on  Chapter  XI 
Review  Exercises  on  Chapter  XII 
Review  Exercises  on  Chapter  XIII 
Review  Exercises  on  Chapter  XIV 
Review  Exercises  on  Chapter  XV 
Review  Exercises  on  Chapter  XVI 
Review  Exercises  on  Chapter  XVII 
Review  Exercises  on  Chapter  XVIII 
Review  Exercises  on  Chapter  XIX 
Review  Exercises  on  Chapter  XX 
Review  Exercises  on  Chapter  XXI 
Miscellaneous  Review  Exercises 
Problems  on  Motion     . 
Problems  involving  the  Lever 
Problems  involving  Geometry 
Miscellaneous  Problems 


327 
328 
329 
330 
332 
333 
334 
335 
336 
337 
338 
339 
340 
343 
345 
347 
349 


LIST   OF   PRINCIPLES 

NUMBER 

I.  Addition  or  Subtraction  of  Numbers  having  a  Common  Factor 

II.  Multiplication  of  the  Sum  or  Difference  of  Two  Numbers 

III.  Division  of  the  Sum  or  Difference  of  Two  Numbers 

IV.  Multiplication  of  the  Product  of  Several  Factors 
V.  Division  of  the  Product  of  Several  Factors 

VI.  Deducing  One  Equation  from  Another 

VII.  Addition  of  Signed  Numbers 

VIII.  Subtraction  of  Signed  Numbers 

IX.  Multiplication  of  Signed  Numbers 

X.  Division  of  Signed  Numbers 

XI.  Arrangement  and  Grouping  of  Terms 

XII.  Removal  and  Insertion  of  Parentheses 

XIII.  Arrangement  and  Grouping  of  Factors 

XIV.  Multiplication  of  Polynomials 
XV.  Product  of  Two  Powers  of  the  Same  Base 

XVI.  Quotient  of  Two  Powers  of  the  Same  Base 

XVII.  Multiplying  or  Dividing  the  Terms  of  a  Fraction 

XVIII.  Square  Root  of  a  Product      .... 


9 
13 
15 
17 
18 
29 
47 
51 
57 
59 
69 
75 
84 
89 
94 
97 
161 
242 


HISTORICAL   NOTES 


Origin  of  the  Arabic,  or  Hindu  Numerals    . 
Origin  of  the  Symbols  of  Operation 
Parentheses  and  Brackets  .... 
Distributive  Laws  of  Multiplication  and  Division 
Origin  of  the  Name  "  Algebra  " 
Representation  of  Unknown  Numbers 
Development  of  Negative  Numbers    . 
Associative  and  Commutative  Laws  of  Addition 
Associative  and  Commutative  Laws  of  Factors 
Exponents  ..... 
The  Pythagorean  Theorem 
Fractional  Notation    . 
Ratio  and  Proportion  . 
Literal  Equations;   Formulas 
Graphic  Representation  of  Equations 
Radicals      ..... 
Quadratic  Equations.     Imaginary  Roots 
The  Binomial  Formula 


PAGE 

5 

11 

15 

32 

41 

62 

69 

84 

101 

151 

161 

188 

206 

239 

242 

291 

313 


PORTRAITS 


Roger  Bacon Frontispiece 

FACING    PACE 

Johann  Widmann       ..........  8 

Franciscus  Vieta         ..........       42 

Sir  William  R.  Hamilton 70 

Sir  Isaac  Newton         .         .  .         .         .         .         .         .         .  .102 

Pythagoras  .         .         .         .         .         .         .         .         .         .         .152 

John  Wallis 188 

Ren6  Descartes 240 

Karl  Friedrich  Gauss 292 

Blaise  Pascal 314 

z 


ELEMENTARY  ALGEBRA 

CHAPTER   I 

INTRODUCTION  TO  ALGEBRA 

LETTERS   USED   TO   REPRESENT   NUMBERS 

1.  Algebra  compared  with  Arithmetic.  —  Algebra,  like  arith- 
metic, deals  with  numbers.  In  arithmetic  numbers  are  repre- 
sented by  means  of  the  Arabic  numerals  C,  1,  2,  3, 4,  5,  6,  7,  8,  9. 
In  algebra  letters,  as  well  as  numerals,  are  used  to  represent 
numbers. 

To  illustrate  this :  If  a  blackboard  is  12  feet  long  and  3 
feet  wide,  its  area  is  12  x  3  =  36  square  feet. 

In  arithmetic  we  write  the  rule : 

area  =  length  X  ividth. 

In  algebra  this  rule  is  abbreviated  to 

^^^^^=  a  =  /xiv, 

where  I  and  w  represent  the  number  of  units  in  the  length  and 
width  respectively,  and  a  the  number  of  square  units  in  the  area. 
But  in  algebra  the  sign  x  is  usually  omitted  between  two 
letters,  so  that  this  rule  is  written : 

a  =  /w. 

Formula.     The   expression,  a  =  Iw,  is  called  a  formula. 
A  formula  is  a  rule  expressed  in  algebraic  symbols. 

Thus,  a  =  Iw  is  a,  formula  giving  the  area  of  a  rfectangle. 

So  we  see  that  one  advantage  in  using  letters  to  represent 
numbers  is  to  shorten  the  rules  of  arithmetic. 

1 


INTRODUCTION   TO   ALGEBRA 


2.    The  systematic  use  of  letters  to  represent  numbers  is  one  of 

the  chief  points  of  difference  between  algebra  and  arithmetic. 
In  the  following  oral  and  written  exercises  note  the  simplicity 
with  which  the  rules  of  arithmetic  may  be  stated  as  formulas 
by  means  of  letters. 

ORAL   EXERCISES 

1.  How  many  square  inches  in  a  slate  8  inches  wide  and 
10  inches  long  ?  In  one  6  inches  wide  and  9  inches  long  ?  In 
one  10  inches  wide  and  I  inches  long  ? 

2.  How  many  square  feet  in  the  top  of  a  table  3  feet  wide 
and  5  feet  long  ?  In  one  2  feet  wide  and  3  feet  long  ?  In  one 
a  feet  wide  and  h  feet  long  ? 

In  arithmetic  we  learned  that  the 
area  of  a  triangle  is  one  half  the  prod- 
uct of  the  base  and  the  altitude,  or 
height.  Thus,  the  area  of  a  triangle  is 
represented  by  the  formula: 

3.  How  many  square  inches  in  a  triangle  whose  base  is  10 
inches  and  whose  altitude  is  4  inches  ?  In  one  whose  base  is 
8  inches  and  altitude  6  inches  ? 

In  arithmetic  we  learned  that  the  volume  of  a  rectangular 
solid  is  the  product  of  the  length,  width,  and  height ;  that  is, 

V  =  Iwh. 

4.  How  many  cubic  inches  in  a  box  3 
inches  wide,  5  inches  long,  and  2  inches 
high  ?  In  one  4  inches  wide,  5  inches 
long,  and  3  inches  high  ?  In  one  w  inches 
wide,  I  inches  long,  and  h  inches  high  ? 

5.  How  many  cubic  yards  in  an  excavation  15  yards  long, 
10  yards  wide,  and  2  yards  deep? 


LETTERS   USED   TO   REPRESENT  NUMBERS  3 

3.  Symbols  of  Operation.  The  signs  of  operation,  +,  — ,  X; 
-i-,  and  the  sign  of  equality,  =,  are  used  in  algebra  with  the 
same  meaning  that  they  have  in  arithmetic.  However,  multi- 
plication is  often  indicated  by  a  point  written  above  the  line  or 
by  omitting  the  sign  altogether,  as  in  the  preceding  formulas. 

Thus,  2  X  I  X  ic  may  be  written  2  •  I  •  lo  or  2  ho. 

Here  letters  differ  from  numerals  ;  for  25  means  20  +  o,  not  2  •  5. 

WRITTEN  EXERCISES 

1.  Write  a  formula  giving  the  area  of  a  rectangle.  State 
this  formula  in  words. 

2.  Using  the  formula  a  =  he,  lind  the  area  in  square  rods 
of  a  field  35  rods  long  and  25  rods  wide. 

3.  Using  the  same  formula,  find  the  area  of  a  rectangle  17 
feet  wide  and  20  feet  long. 

4.  Using  the  formula  «  =  i  bh,  find  the  area  of  a  triangle 
whose  base  is  16  feet  and  whose  altitude  is  11  feet.  State  the 
process  in  words. 

5.  Using  the  same  formula,  find  the  area  of  a  triangle  whose 
base  is  37  rods  and  whose  altitude  is  23  rods. 

6.  Given  the  length,  width,  and  height  of  a  rectangular 
solid,  how  do  you  find  its  volume  ?  State  this  rule  as  a 
formula,  using  Z,  w,  h,  and  v  to  represent  the  length,  width, 
height,  and  volume,  respectively.     State  it  also  in  words. 

7.  Using  the  formula  v  =  lirh,  find  the  volume  of  a  coal 
bin  15  feet  long,  12  feet  wide,  and  9  feet  high. 

8.  How  many  cubic  feet  will  a  freight  car  hold  which  is 
40  feet  long,  8  feet  wide,  and  7  feet  high  ? 

9.  An  excavation  for  a  basement  is  45  feet  long,  28  feet 
wide,  and  8  feet  deep.     How  many  cubic  feet  in  it  ? 


4  INTRODUCTION   TO   ALGEBRA 

WRITTEN  EXERCISES 

By  arithmetic  8  per  cent  of  90  means  .08  x  90  =  7.2.  In  this 
example  90  is  called  the  base,  .08  the  rate,  and  7.2  the  percentage. 

1.  In  like  manner  find  6  per  cent  of  125 ;  5  per  cent  of  350 ;  3 
per  cent  of  80 ;  and  10  per  cent  of  4.9.  In  each  case  how  is 
the  percentage  found  ? 

The  rule  for  finding  the  percentage  is  stated  in  the  following 
formula,  in  which  b  represents  the  base,  r  the  rate,  and  p  the 
percentage  :  p  =  br. 

2.  Using  the  formula  p  =  br,  find  the  percentage  if  6  =  64.8, 
r  =  16%  ;  also  if  6  =  1080,  r  =  7  %. 

By  a  rule  of  arithmetic  the  interest  on  $  500  for  3  years 
at  6  per  cent  is  500  x  .06  x  3  =  90.  In  this  example  500  is 
the  principal,  .06  the  rate,  and  3  the  time. 

3.  In  like  manner,  find  the  interest  on  $900  for  7  years  at  4 
per  cent,  on  $  1250  for  5  years  at  5  per  cent,  on  $  800  for  10 
years  at  6  per  cent. 

The  rule  for  finding  the  interest  is  stated  in  the  following 
formula,  in  which  p  represents  the  principal,  r  the  rate,  t  the 
time,  and  i  the  interest :       f  —ppf^ 

4.  Using  the  formula,  i  =  prt,  find  the  interest  on  $  1500 
for  12  years  at  6  per  cent.  Also  find  i  if  p  =  1750,  ?•  =  5  %, 
and  t  =  4:. 

5.  Find  the  area  of  a  rectangle  24  feet  long  and  18  feet 
wide.     State  the  formula  used. 

6.  Find  the  volume  of  a  rectangular  coal  bin  14  feet  long, 
10  feet  wide,  and  6  feet  deep.     State  the  formula  used. 

7.  Find  the  percentage  if  the  base  1j  74  and  the  rate  14 
per  cent.     State  the  formula  used. 

8.  Find  the  interest  if  the  principal  is  $  840,  the  rate  6 
per  cent,  and  the  time  3  years.     State  the  formula  used. 


LETTERS  USED   TO  REPRESENT   NUMBERS  5 

HISTORICAL  NOTE 

The  Arabic  or  Hindu  Numerals.  The  Arabs  brought  our  present  sys- 
tem of  numerals  to  Europe  when  they  invaded  Spain  from  Morocco 
in  the  eighth  century  of  our  era.  It  is  now  known  that  these  numerals, 
including  zero,  are  really  of  Hindu  origin  and  that  they  were  invented 
between  the  years  200  and  650  a.d. 

In  the  eleventh  and  twelfth  centuries  the  Mohammedan  Arabs,  who 
had  settled  in  Spain  and  Northern  Africa,  traded  with  the  merchants 
from  the  great  Italian  cities,  and  in  this  way  tlieir  system  of  numerals 
gradually  spread  over  Europe. 

In  1202  Leonardo  Fibonacci  of  Pisa  wrote  a  book  on  mathematics,  the 
Liber  Abaci.,  in  wliich  he  explained  the  Arabic  numerals  for  the  first  time 
to  the  people  of  Italy.  He  says  that  he  published  this  book  "  in  order 
that  the  Latin  (Italian)  race  might  no  longer  be  deficient  in  that  knowl- 
edge." The  use  of  the  new  system  spread  but  slowly,  however,  and  it 
was  not  until  the  seventeenth  century  that  it  came  to  be  used  exclusively 
in  Germany  and  England. 

The  Greeks  and  Romans  had  used  letters  for  number  symbols,  but  the 
great  step  in  advance  was  taken  by  the  Hindus  when  they  introduced  the 
principle  of  place  value  which  was  made  possible  by  the  symbol  for  zero. 
Thus,  in  51  =  50  +  1,  the  6  means  5  tens,  while  in  VI  =  V  -|-  1,  the  V 
means  simply  five  ones. 

Of  all  mathematical  inventions  none  has  contributed  more  to  general 
intelligence  and  material  welfare  than  this  one.  To  appreciate  this,  one 
needs  only  to  try  to  multiply  two  such  numbers  as  589  and  642  when 
expressed  in  the  Roman  notation  ;  that  is,  to  multiply  DLXXXIX  by 
DCXLII.  This  difficulty  alone  would  account  for  the  very  slow  develop- 
ment of  algebra  until  better  symbols  were  invented. 

Indian  Numerals  about  950,  a.d.|     \7^^\S^      /     \,  [^ 

Arabic  Numerals  about  1100,  a.d.  \    ^      ^   Q^   U     V     0    9    3^' 
Of  German  origin  about  1385,  A.D.  ^    ->     'j    O     /     ^^  \    5^    C\     7 /A' 
Probably  Italian  about  1400,  a.d.  17o3^^o'7     SOI^ 


6  INTRODUCTION  TO   ALGEBRA 

ALGEBRAIC  EXPRESSIONS 
4.    Algebraic  Expressions.     We   have   seen   that   in   algebra 
letters  as  well  as  numerals   are   used  to  represent   numbers. 
We  have  seen  also  that  the  same  signs  of  operation  are  used  as 
r  in  arithmetic. 

Any  combination  of  numerals,  letters,  and  signs  of  operation, 
used  for  the  purpose  of  representing  numbers,  is  called  nii 
algebraic  expression. 

E.g.  38,  18  r,  5  n  +  8  n,  3  a  —  2  are  algebraic  expressions. 

Expressions  such  as  5  n  +  8  n  and  3  a  —  2  are  said  to  be 
written  in  symbols ;  that  is,  by  means  of  numerals,  letters,  and 
signs  of  operation. 

The  expression  3a  — 2  written  in  words  would  be  "three  times  the 
number  a  diminished  by  2."  See  how  much  simpler  the  algebraic  expres- 
sion is. 

ORAL  EXERCISES 

Read  the  expressions  in  Examples  1  to  7. 

1.  3  +  5 ;  a  +  h]  3  ^'  -f-  2  ?i ;  2  A:  -f-  ^? ;  li  +  2  n. 

2.  7  -  3  ;  a  -  6 ;  3  A;  -  2  ?i ;  2  k -n;  k-2n. 

3.  4x7;  3  •  5 ;  a6  ;  ahc ;  4  mn ;  2  a.c ;  3  cd. 

4.  10-^5;  I ;  G  ^  d\  m  -^  n\  2h  -i-  a\  ^n  -r-k. 

5.  3  a;  4a6c;  4-5«6;  3m?i;  3  m— 2  n;  4o&— 2  c. 

6.  2k',  2A:  +  1;  2k-l;  4A:  +  1;  Ak-l. 

7.  f^;  J-y;  ^n;  ^k-,  |  abc ;  2  ab;  3  cd  —  ^  mn. 

8.  If  one  yard  of  wire  costs  3^,  how  many  cents  will  o 
yards  cost  ?  8  yards  ?  7i  yards  ?     State  the  process  in  words. 

9.  If  one  carfare  costs  5  cents,  how  much  will  10  carfares 
cost  ?  n  carfares  ? 

10.  How  many  days  in  3  weeks  ?  in  n  weeks  ? 

11.  How  many  feet  in  4  yards?  in  n  yards  ? 

12.  How  many  inches  in  12  feet  ?  in  20  feet  ?  in  k  feet? 

13.  I  low  many  cents  in  $1  ?  in  S5  ?  in  S  ?i  ? 


ALGEBRAIC   EXPRESSIONS  7 

WRITTEN  EXERCISES 

1.  If  a  and  h  are  numbers,  express  in  symbols  their  sum 
and  also  their  product. 

2.  If  m  and  n  are  numbers,  write  in  symbols  m  divided  by 
n]  also  771  minus  n. 

3.  If  p  and  q  are  numbers,  write  the  sum  of  5  times  p  and 
3  times  q. 

4.  If  a,  b,  and  c  are  numbers,  write  in  symbols  that  a  multi- 
plied by  b  equals  c ;  also  that  c  divided  by  a  equals  b. 

If  a  =  3,  6  =  5,  c  =  7,  hnd  the  value  of  each  of  the  following 
algebraic  expressions. 

4c  -3a.  15.  36-  2c +  3. 

5c  +  ofc.  16.  4c  — 6+ 3a. 

ab  —  c  +  2.  17.  abc  —  a  —  b. 

4  a  —  2  6  4-  3  c.  18.  ab  -}-  be -{-  ac. 

14.    2  a  -\-  ab  -\-  7  c.  19.  ac  -\-bc  —  ab. 

20.  If  )ii,  n,  and  p  are  numbers,  write  5  times  the  product  of 
m  and  n,  minus  3  times  the  product  of  m  and  p. 

HISTORICAL  NOTE 

Origin  of  the  Symbols  of  Operation.  It  was  about  the  year  1500  a.d. 
that  our  present  symbols  indicating  addition  and  subtraction  first  ap- 
peared in  a  book  by  a  German  named  Johann  Widmann.  The  sign  x 
for  multiplication  was  first  used  about  50  years  later  by  an  Englishman, 
William  Oughtred.  About  the  same  time  the  sign  =  was  first  used  by 
Robert  Recorde,  also  an  Englishman  ;  but  the  sign  -^  for  division  does  not 
appear  until  1659,  when  it  was  used  by  a  German,  J.  IT.  Rahn.  It  should 
not  be  imagined,  however,  that  any  one  of  these  symbols  came  suddenly 
into  general  use.     A  distinguished  historian  of  mathematics  has  said  : 

"  Our  present  notation  has  arisen  by  almost  insensible  gradations  as 
convenience  suggested  different  marks  of  abbreviation  to  different  authors  ; 
and  that  perfect  symbolic  language  which  addresses  itself  solely  to  the 
eye,  and  enables  us  to  take  in  at  a  glance  the  most  complicated  relations  of 
quantity,  is  the  result  of  a  long  series  of  small  improvements." 


5. 

a  +  b  -\-  c. 

10 

6. 

a  -\-b  —  c. 

11 

7. 

2  a  —  6  +  c. 

12 

8. 

56-2c. 

13 

9. 

1&  a  —  Q  c. 

14 

8  INTRODUCTION   TO   ALGEBRA 

5.  Factors.  Many  algebraic  expressions,  such  as  abc,  2  mn, 
Sxy,  represent  products.  Numbers  which  multiplied  to- 
gether form  a  product  are  called  factors  of  that  product. 

E.g.  3  and  4  are  factors  of  12,  as  are  also  2  and  6,  1  and  12.  a,  b,  c, 
are  factors  of  abc,  as  are  also  a  and  be,  b  and  ac,  c  and  ab. 

Coefficients,  If  an  expression  is  the  product  of  two  factors, 
then  either  of  these  factors  is  called  the  coefUcient  of  the  other. 

E.g.  In  9  rt,  9  is  the  coefficient  of  rt,  r  is  the  coefficient  of  9 1,  and  t  is 
the  coefficient  of  9  r.  But  in  such  an  expression  the  factor  represented  by 
Arabic  numerals  is  usually  regarded  as  the  coefficient. 

ADDITION  AND  SUBTRACTION   OF  NUMBERS   HAVING  A  COMMON 

FACTOR 

6.    Addition.     Two  numbers  which  have  a  common  factor, 
such  as  5  •  3  and  8  •  3,  may  be  added  in  two  ways. 

By  the  first  method,  ive  perform  the  iyidicated  multiplications 
and  then  add  the  products. 

Thus  5  .  3  +  8  ■  3  =  15  +  24  =  39. 

By  the  second  method,  we  add  the  coefficients  of  the  common 
factor  and  then  multiply  this  sum  by  the  common  factor. 

Thus,  5  •  3  +  8  •  3  =  13  .  3  =  39. 

This  is  evident,  since  5  times  a  number  and  8  times  the  same  number 
make  13  times  that  number. 

In  case  the  common  factor  is  represented  by  a  letter,  the 
second  process  only  is  available. 

Thus  5  n  +  8  7i  =  13  n. 

ORAL  EXERCISES 

In  the  manner  shown  above  add  the  following. 

1.  2a;  +  4x.  4.    3a +  2  a.  1.    ^t  +  7  t  +  ^t. 

2.  4  a  +  3  a.  5.    5  •  6  +  3  •  6.  8.    3  r  +  5  r  +  7  r. 

3.  13  7i-h4n.        6.    "ih  +  ^b  +  h.        9.    5.9  +  4.9  +  6.9. 


Johann  Widmann  was  born  at  Eger  in  Austria  in  1460.     He 

matriculated  at  Leipsic  in  1480  and  later  became  a  lecturer  there 
on  algebra.  In  1489  he  wrote  a  treatise  on  mercantile  arith- 
metic, in  which  are  first  found  the  symbols  +  and  — ,  used  as 
marks  of  excess  and  deficiency. 

Widmann  is  chiefly  interesting  to  us  as  the  first  German 
scholar  whose  name  is  associated  with  the  subject  of  algebra. 
He  has  been  called  the  originator  of  German  algebra,  though 
Christoff  Rudoiff,  a  pupil  of  Widmann's,  was  the  first  to  write 
a  textbook  of  algebra  in  the  German  language.  Widmann  was 
the  author  of  a  book  on  geometry. 


ADDITION   AND   SUBTRACTION  9 

7.  Subtraction.  Just  as  we  add  two  numbers  having  a  com- 
mon factor  by  adding  the  coefficients  of  the  common  factor, 
so  we  subtract  two  such  numbers  by  subtracting  the  coefficients. 

Thus,  from  64  =  8  •  8  From  84  =  12  •  7  From  17  n 

subtract  48  =  6  •  8  subtract  49  =    7-7  subtract    6  n 

Remainder  16  =  2  •  8  35  =    5-7  Tin 

ORAL  EXERCISES 

In  this  way  perform  the  following  subtractions : 

1.  8.7-3.7.            5.    10  b- 4.  b.  9.  14  a- 8a. 

2.  6.99-5.99.        6.      7  a- 4  a.  10.  12  6-9  6. 

3.  6n-2  7i.                7.    23x-16x.  11.  8^-2^. 

4.  8  a  — 3  a.                 8.    15  71  — 3  n.  12.  19  r  -  11  r. 
The  foregoing  examples  illustrate 

Principle  I 

8.  Rule.  To  find  tlie  sum  of  two  numbers  having  a 
common  factor,  add  the  coefficients  of  the  coimnon  factor 
and  multiply  the  result  hy  the  common  factor. 

To  find  the  difference  between  two  numbers  having  a 
common  factor,  subtract  the  coefficients  of  the  cominon 
factor  and  multiply  the  result  by  the  common  factor. 

ORAL  EXERCISES 

Perform  the  following  indicated  operations : 


1. 

5x-\-l  X. 

9. 

4  xy  -^Q  xy  -\-4:xy. 

2. 

^x-3x. 

10. 

2  a.r  -h  3  arc  —  4  ax. 

3. 

9  a  +  8  a. 

11. 

G  .r  +  l-lx—lx. 

4. 

2  m  +  6  m. 

12. 

20x-Sx-3x. 

5. 

82/  +  52/- 

7 

y- 

13. 

5  by  -{-7  by -3  by. 

6. 

3a+7a- 

6 

a. 

14. 

4  a6c  +  3  abc  +  2  abc. 

7. 

5  a6  +  7  ah 

■3ab. 

15. 

6  ay  -\- 9  ay  —  3  ay. 

8. 

3x+^x— 

8 

X. 

16. 

12  x -\- S  X -{- 4.  x. 

10  INTRODUCTION   TO   ALGEBRA 

9.  Checking  Results.  The  substitution  of  numerical  values 
for  letters  in  order  to  test  the  correctness  of  an  operation  is 
called  checking  the  result. 

E.g.     To  check  2x+  7a;-f4a;  =  13x,  we  may  substitute  2  for  x. 
Thus,  2-2  +  7. 2  +  4. 2=4  + 14 +  8  =  26  =  13.  2. 

WRITTEN  EXERCISES 

Perform  the  following  indicated  operations  and  check  the 
results  in  the  first  eight  by  letting  t  =  l,  n  =  2,  x  =  lj  A:=l, 
s=l,  a  =  1. 

1.  68^  —  11^.  13.  17  St -hS  St  — 12  St. 

2.  15?i  +  2on  — 18/?.  14.  12  abc  —  2  abc  -  6  abc. 

3.  70.T- 15a;  +  7a;  — 23  a;.  15.  42  .t?/ +  6  a;?/ —  35  a:?/. 

4.  4.Sk-3k-2k-\-6k.  16.  29  rst  -  IS  rst  -  6  rst. 

5.  207i-16n-{-2n.  17.  32  oc- 17  ac  + 2  ac. 

6.  25^  +  20^-3^.  18.  91a-81a  +  2a. 

7.  28s-3s  +  20s,  19.  10  a; +  24  a;  +  8a; -40  a;. 

8.  36  a  — 14a  +  3a  — 2a.  20.  5  ?/ +  31?/  — 9  ?/— 21  ?/. 

9.  7  a6  —  3  a?>  +-  2  ab.  21.  63  c  -  47  c  —  8  c  +  7  c. 

10.  34:  rs -  12  rs- 5  rs.  22.  16^-11^-2^  +  3^. 

11.  S4:X?j  —  18  xy  —  4  xy.  23.  12  xy  —  9xy-\-S  xy. 

12.  lSmyi  —  lmn  —  3mn.  24.  39  a6  —  27  a6  —  8  a&. 

25.  3  rs«  +  9  Tst  +  26  rst  —  18  rst. 

26.  47  abc  —  14  abc  —  8  abc  +  3  abc. 

27.  31  xyz  +  42  xyz  —  39  a;_y2;  +  7  a;?/2;  +  17  a;?/^. 

28.  3  mn  +  24  mn  —  14  ??i/i  +  8  mn  +  11 7>i7i. 

29.  78  qr  —  13  qr  +  8  gr  +  7  gr  —  12  qr. 

30.  12  /w  +  7  Z/;i  —  9  /?/i  +  46  Im  —  7  Im. 

31.  145  a;  +  17  .a;  -  125  a;  +  280  .T. 

32.  40  kl  +  260  A:?  -  34  kl  -  79  A-L 

33.  7  ?/  +  14  y  +  28  ?/  +  56  ?/  +  112  ?/. 

34.  93  2  +  47  2  +  82  ;2  +  235  z  -  406  z. 


ADDITION   AND   SUBTRACTION  11 

10.  Equations  and  Identities.  Two  algebraic  expressions  rep- 
resenting the  same  number  and  connected  by  the  sign  =  form 
an  equation,  as  7i  +3  =  5. 

The  expressions  thus  connected  are  called  the  members  of  the 
equation.  In  the  above  equation,  n  -{-  3  is  the  left  member,  and 
5  is  the  right  member. 

In  general,  an  equation  is  true  only  when  the  letters  have 
certain  special  values.     Thus  ?i  +  3  =  5  is  true  only  when  n  =  2. 

The  equation  5n  +  8n  =  13  7i  is  true  for  all  values  of  n. 
Such  an  equation  is  called  an  identity. 

When  it  is  desired  to  emphasize  that  an  equation  is  an  iden- 
tity, the  sign  =  is  used.     That  is,  3  a  +  5  a  =  8  a. 

An  equation  such  as  3  x  4  =  12,  in  which  both  members  are 
expressed  in  Arabic  numerals  is  also  called  an  identity. 

11.  Symbols  of  Aggregation.  Parentheses  are  used  to  indicate 
that  some  operation  is  to  be  extended  over  the  whole  expres- 
sion inclosed  by  them.  Thus  2{x-\-y)  means  that  the  sum  of 
X  and  y  is  to  be  multiplied  by  2,  while  2x-^y  means  that  x 
alone  is  to  be  multiplied  by  2. 

Instead  of  parentheses,  brackets  [  ],  braces  \  |,  or  the  vin- 
culum ,  may  be  used  with  the  same  meaning.  All  such 
symbols  are  called  symbols  of  aggregation. 


E.g.    2{x  ■\-  ?/),  2[x  +  ?/],  2{x  +  y},  2x  +  y  all  mean  the  same  thing. 

ORAL  EXERCISES 

Evaluate  (find  the  value  of)  each  of  the  following  when 
a  =  4,  h  =  2,  c  =  l,  x  =  4:,  y  =  6. 

1.  3(a+&).  ^.    x-\-y-^2.  7.   2[x  +  y-S']. 

2.  4(a  — &).  5.    4  a  — 6.  8.    Sla  +  b  +  c^. 

3.  (x-\-y)-^2.  '6.    oa  +  b.  9.    S(a -\- b) -\- c. 

Historical  Note.  Parentheses  (  )  were  first  used  with  their  present 
meaning  by  an  Englishman,  A.  Girard,  in  a  book  on  "  Arithmetic,"  pub- 
lished in  the  year  1629.  Brackets  and  braces  are  of  later  origin,  as  is  also 
the  sign  =  to  denote  identity . 


12  INTRODUCTION   TO   ALGEBRA 

MULTIPLICATION  OF  THE  SUM  OR  DIFFERENCE  OF  TWO  NUMBERS 

12.  Multiplying  a  Sum.  The  sum  of  two  or  more  arith- 
metical numbers  may  be  multiplied  by  another  number  in  two 
ways. 

By  the  first  method  ive  add  the  numbers  and  then  multiply 
their  sum  by  the  other  number. 

Thus  4(2  +  7)  =  4  .  9  =  36  ; 

3(3  +  8  4-  9)  =  3  .  20  =  60. 

By  the  second  method  we  multiply  each  of  the  numbers  sepor 
rately  and  then  add  the  products. 

Thus  4(2 +  7)  =  4.2 +4 -7  =8  +  28  =  36; 

3(3  +  8  +  9)  =  3  .  3  +  3  •  8  +  3  .  9  =  9  +  24  +  27  =  60. 

But  in  case  the  numbers  are  represented  by  letters,  the  second 
process  only  is  available. 

E.g.    3(a  +  6)=3a4-3&  and  m{r  +  s)  =  mr  +  ms. 

ORAL  EXERCISES 

Multiply  each  of  the  following  in  two  ways  when  possible  : 

1.  3(2  +  4).  7.  3(a  +  6).  13.    7(4  +  ?/ +  3). 

2.  4(3  +  5).  8.  ll{h  +  k).  14.    6(2  + a +  6). 

3.  2(4  +  6).  9.  4(a  +  5  +  c).  15.   3(2  +  3  +  4). 

4.  5(3  +  2).  10.  7{x  +  y).  16.   4(a  +  0  +  6). 

5.  4(7  +  3).  11.  5(a;  +  2/  +  3).  17.   4(6 +  3)+ 2. 

6.  8(5  +  1).  12.  4(a  +  6  +  l).  18.    2(6 +  6) +3. 

13.  Multiplying  a  Difference.  The  difference  of  two  arith- 
metical numbers  may  likewise  be  multiplied  by  a  given  number 
in  either  of  two  ways. 

E.g.  6(8 -3)  =  6.  6  =  30, 

or  6(8 -3)  =  (6.  8) -(6.  3)  =48- 18  =30. 

But  in  case  the  numbers  are  represented  by  letters,  the 
second  process  only  is  available. 

E.g.  (S{r  —  t)  =  Qr  —  (St  and  a{c  —  d)  =  ac  —  ad. 


MULTIPLICATION   OF  A   SUM  OR  DIFFERENCE  13 


ORAL  EXERCISES 

Perform  as  many  as  possible  of  the 

following  multiplications 

in  two  ways : 

1.    7(5-2).                  6.    8(/i-4). 

11.    x(y-z). 

2.    12(7-4).                7.    5(x-l). 

12.    t{u  —  v). 

3.    5(10-8).               8.   S(y-2). 

13.    x{y-S). 

4.    7(6-3).                 9.    a(c-d). 

14.    a(x-7). 

5.    9(a_2).               10.    m(r  —  s). 

15.    6(2-4). 

The  foregoing  examples  illustrate 

Principle  II 

14.  Rule.  To  multiply  the  sum  of  two  numbers  by  a 
given  number,  multiply  each  of  the  numbers  separately 
by  the  given  number,  and  add  tlie  products. 

To  m^ultiply  the  difference  of  two  numbers  by  a  given 
number,  multiply  each  of  the  numbers  separately  by  tJie 
given  number,  and  subtract  the  products. 

WRITTEN  EXERCISES 

1.  Multiply  5-1-7  +  11  by  3  without  first  adding,  and  then 
check  by  performing  the  addition  before  multiplying, 

2.  Multiply  m  -h  w  by  4  and  check  for  m  =  5,  n  =  7. 

Solution,  4(m  +  ?i)  =  4  m  +  4  n. 

Check.  4(5  +  7)   =  4  •  12  =  48,  also 

4  .  5  +  4  .  7  =  20  +  28  =  48. 

3.  Multiply  x-{-  y  by  r  and  check  for  x  =  2,  y  —  ^,  r  =  Q>. 

4.  Multiply  r  -\-  shy  k  and  check  for  ?*  =  4,  s  =  5,  k  =  6. 

5.  Multiply  a-[-h\)j  7n  and  check  for  a  =  3,  6  =  2,  m  =  4. 

6.  Multiply  m  —  n  -h  2  by  c  and  check  for  m  =  5,  n  =  2,  and 
c  =  3. 

7.  Multiply  a  —  5  —  c  by  fZ  and  check  for  a  =  10,  6  =  3,  c  =  4, 
and  d  =  8. 


14 


INTRODUCTION  TO   ALGEBRA 


DIVISION   OF  THE  SUM  OR  DIFFERENCE  OF   TWO  NUMBERS 

15.  Dividing  a  Sum  or  a  Difference.  In  dividing  the  snm  or 
difference  of  two  arithmetical  numbers  by  a  given  number,  the 
process  may  be  carried  out  in  two  ways. 

By  the  first  method  : 

(12 +  8) -=-2  =  20 --2  =  10; 
(20-  12)-4  =  8h-4  =  2. 

By  the  second  method : 

(12  +  8)-2=(12-2)  +  (8--2)=6  +  4  =  10; 
(20-  12) -f- 4  =  (20-f-4)-(12^4)=o-3  =  2.  . 

If  the  numbers  in  the  dividend  are  represented  by  letters, 
then  usually  the  second  method  only  is  available. 
E.g.     (r  +  0-  5  =(r  -  5)+  {t  -  5). 

Division  in  algebra  is  often  indicated  by  the  fractional  form. 
Thus,  {r  -[- 1)  -^  b  ={r  -i-  b)  ■\-  {t  -^  b)  is  cornmouly  wi'itten 

r  +  t  _r      t 
5     ~5      5* 

In  either  case  this  is  read :  r  plus  t  divided  by  5  equals  r 
divided  by  5  plus  t  divided  by  5. 


ORAL  EXERCISES 

Perform  each  of  the  following  divisions 

in  two  ways  when 

this  is  possible : 

1.    (4 +  12)- 4.           6.    (12-6)-T-3. 

11.    {x  +  y)^4.. 

2.    (8 +  4)- 2.            7.    (18 -12)- 6. 

12.    (m +  ?•)-=- 9. 

3.    (10f5)-5.           8.    (21 -14) -7. 

13.    (m  — r)H-9. 

4.    (9^  12)h-3.          9.    {x-y)^2. 

14.    (/•-?)  — 5. 

5.    (16-f8)--4.         10.    {y-z)^(S. 

15.    (rH-^)  — 5. 

16.    (8-f6-4)H-2.               20.    (tt 

■f  4  +  6)H-2. 

17.    (9 +  0-3)- 3.                21.    (8- 

-b-c)^4.. 

18.    (12-4-|-8)h-4.               22.    (94-c  +  d)H-3. 

19.    {a-{-h-  c)-i-2.                  23.    (a: 

-2/ +  12) -6. 

DIVISION   OF   A   SUM   OR   DIFFERENCE  15 

The  foregoing  examples  illustrate 

Principle  III 

16.  Rule.  To  divide  the  sum  of  two  nuinhers  by  a  given 
nuniber,  divide  each  number  separately  and  add  tJie 
quotients. 

To  divide  the  difference  of  two  numbers  by  a  given 
number,  divide  each  number  separately  and  subtract  tlie 
quotients. 

WRITTEN   EXERCISES 

1.  Divide  72  +  56  by  8  without  first  adding. 

2.  Divide  144  —  36  by  12  without  first  subtracting. 

3.  Divide  r-{-t  by  5  and  check  the  quotient  when  r  =  15, 
« =  25 ;  also  when  r  =  60,  ^  =  30. 

4.  Multiply  7  +  9  by  3  without  first  adding  7  and  9. 

5.  Find  the  product  of  12  and  a-\-h,  checking  the  result 
when  a  —  h,  h  —  1.  . 

Perform  the  following  indicated  operations  : 

6.  S{a  +  h  +  G  +  d).     Check  for  a  =  1,  /j  =  2,  c  =  3,  d  =  4. 

7.  7(r  —  s-\-t  —  x).     Check  for  r  =  t  =  b,  s  =  a;  =  4. 

8.  (7>i  +  7i  +  r)--4.     Check  f or  m  ==  64,  n  =  32,  r  =  8. 

9.  {x-\-y-\-z)^^.     Check  for  x  =  15,  y  =  10,  and  z=z5. 

10.  74  rs  -  67  rs  -  2  rs- 3  rs.         12.    a(4  -  cZ  +  6  +  c  + 3). 

11.  (63-3o-14  +  21)--7.  13.    (;21-x-y-^3  +  c)k. 

HISTORICAL  irOTE 

Fundamental  Laws  for  Multiplication  and  Division.  The  fundamental 
character  of  Principles  II  and  III  was  not  fully  appreciated  until  the  first 
part  of  the  last  century.  Principle  II  states  what  is  called  the  Distributive 
Law  of  Multiplication  with  respect  to  addition  and  subtraction.  That  is, 
the  multiplier  is  distrilmted  over  the  multiplicand.  The  name  was  first 
used  by  a  Frenchman,  F.  J.  Servois,  in  a  paper  published  in  1814.  Prin- 
ciple III  states  the  Distributive  Law  of  Division  with  respect  to  addition 
and  subtraction.     Compare  notes  on  pages  69  and  84. 


16  INTRODUCTION   TO  ALGEBRA 

MISCELLANEOUS  EXERCISES. 

1.  (48 +  36 +  24 +  12)- 6. 

2.  (35 -20 +  30 -40)-- 5. 

3.  3(a;  +  2/  +  2;)+2(?/-2  +x). 

4.  3(a  +  &  +  c)  +2(a  -  6  +  c). 

5.  5(a;  +  ?/  +  2;)  +  3(aj  -  ?/  +  2;). 

6.  4:(ab  +  ccZ  +  e/)  +2(a6  +  cd  -  ef). 

7.  49pQ'  +  18pg  — 62pg  +  3pg. 

8.  13  xyz  +  3  iKi/2;  —  8  xyz  —  7  xyz. 

9.  {q  +  r-\-s  +  t  —  a—h)^c. 

10.  35  Im  —  33  Zm  +  7  Z??i  —  3  Z?/!  —  2  Im. 

11.  Z:(^  +  ?7i  +  7i  +  r— s  — ^). 

12.  a(c  +  fZ  —  e  +/—  .g). 

13.  27  ahc  —  19  a?>c  —  4  a6c  +  8  a6c. 

14.  (a  +  r  +  s  — ^  — g)-^3. 

15.    For  what  values  of  a,  6,  c,  d  are  the  following  equations 

(a)  a6 +  ac  +  a(Z  =  a(6 +  c  +  d). 
(6)  ah  -\- ao,  —  ad  =.  a{h  +  c  —  d). 

^  ^  d  d     d      d 

MULTIPLICATION   OF   A  PRODUCT 

17.  Multiplying  a  Product.  In  multiplying  a  product  like 
3  •  5  by  2  the  work  may  be  carried  out  in  three  different  ways. 

Thus,  2.  (3.  5)  =2. 15=30; 

or  2.  (3.6)  =  6.5zz30; 

or  2.  (3.  5)  =3. 10  =  30. 

In  multiplying  3  n  by  2  we  could  indicate  the  work  in  these 
three  ways,  but  we  can  perform  it  in  the  second  way  only; 
namely,  2  •  (3  n)  =  6  n. 

We  may  evidently  choose  any  one  of  the  factors  of  the 
multiplicand  and  multiply  this  one  alone  by  the  multiplier. 


MULTIPLICATION   OF   A   PRODUCT 


17 


ORAL  EXERCISES 


Pind  each  product  in  several  ways  and  verify  by  comparing 
results. 


1.  2(2.3). 

2.  2(3-4). 

3.  2(4-5). 

4.  3(3-4). 

5.  2(2  -3.4). 


6.  3(2-3-4). 

7.  2(5-10). 

8.  2(5-6). 

9.  4(5-5). 
10.  3(5-4). 


11.  8(6-2). 

12.  9(3-4.) 

13.  7  (2  -  3). 

14.  10(2-4). 

15.  2(4-5  -2), 


These  examples  illustrate 


Principle  IV 

18.  Rule.  To  multiply  the  product  of  several  factors  hy 
a  given  number,  rnultiply  any  one  of  the  factors  hy  that 
number. 

Principles  IV  and  II  should  be  carefully  contrasted. 


Thus, 


but 


2(2. 3.  5)  =4. 3. 5  =  2. 6. 5zz2. 3- 10 
2(2  +  3  +  5)  =  4  +  6  +  10  =  20. 


60, 


l7i  multiplying  the  product  of  several  numbers  ice  operate  upon 
any  one  of  them,  bat  in  multiplying  the  sum  or  difference  of  num- 
bers lue  operate  upon  each  of  them. 


WRITTEN  EXERCISES 

Multiply  as  many  as  possible  of  the  following   in   two  or 
more  ways. 

8.  15(7  abc). 

9.  15  (Tab). 

10.  33(4m?i). 

11.  47(2x1/). 

12.  12(16  rs). 

13.  6(30  xy). 

14.  9(3-4-5). 


1.  7(3-4-5). 

2.  8(7-2  -3). 

3.  9  (2  -  3  .  4). 

4.  15(2a5). 

5.  lS(5xy). 

6.  23(8x7/2;). 

7.  4(19  -5). 


15. 

7(8 

5). 

16. 

8(9. 

5). 

17. 

3(9 

-2.6). 

18. 

2(4 

7-5). 

19. 

6(5- 

4-6). 

20. 

5(4- 

5  -  x). 

21. 

4(5- 

8  •  y)- 

18  INTRODUCTION  TO   ALGEBRA 

DIVISION   OF   A  PRODUCT 

19.  Dividing  a  Product.  Just  as  in  multiplying  the  product 
of  several  factors  by  a  given  number,  so  also  in  dividing  such 
a  product  by  a  given  number,  we  may  operate  upon  any  one  of 
the  factors  separately. 

E.g.  (4  .  6  .  10)  --  2  =  240  --  2  =  120. 

Also  (4  •  6  .  10)  -  2  =  2  .  6  .  10  =  120, 

(4-6.  10)-f-2  =4.3.10  =  120, 
and  (4  .  6  •  10)  --  2  =  4  .  6  .  5  =  120. 

Note  that  in  each  case  only  one  factor  is  divided. 

ORAL   EXERCISES 

Perform  each  of  the  following  divisions  in  more  than  one 
way  where  possible : 

1.  (5  .  8  .  3)  -r-  2.  8.  14  xyz  -^  x. 

2.  20  abc  -V-  4.  9.  (16  •  18  •  13)  --  8. 

3.  12  abc -^3.  10.  5o7>c-7-a. 

4.  (3.  20.  8) -4- 4.  11.  loxy-^3, 

5.  14:  xyz  ^7.  12.  5  (10  .t  +  15  ?/) -f- 5. 

6.  12abc^c.  13.  8(3aj  -  4?/) --4. 

7.  (10.  5.  3) --5.  14.  G(2s-30-3. 

The  foregoing  examples  illustrate 

Principle  V 

20.  Rule.  To  divide  the  product  of  several  factors  hy 
a  given  number  divide  any  one  of  the  factors  hy  that 
number. 

21.  Cancellation.  Principle  V  is  already  known  in  arithmetic 
in  the  process  called  cancellation. 

Thus,  in  the  fraction  '  '  '  ,  the  3  may  be  cancelled  out  of  either 
6  or  9,  giving  ^'^  "^  =  2  •  2  •  0  or  2  •  (5  •  8. 


DIVISION   OF   A   PRODUCT  19 

Contrast  Principles  III  and  V. 

By  Principle  V,  (4  .6- 8) -2  =  2- 6- 8=4- 3- 8  =  4- 6- 4  =  96." 
By  Principle  III,  (4  +  6  +  8)--2  =  2  +  3  +  4  =  9. 

That  is,  in  dividing  the  product  of  several  numbers  loe  operate 
upon  any  one  of  them,  hut  in  dividing  their  sum  or  difference  we 
operate  upon  each  of  them. 

WRITTEN   EXERCISES 

Perform  the  following  indicated  operations,  remembering 
that  a  fraction  is  merely  an  indicated  division : 

1.  3a(7-c  +  6)--a.  4.   13  (8  -  46 -f- 12a)  --4. 

2.  5  6c  (r«  —  e  +  3)  -7-  he.  5.    14  (7  —  7  m  +  14  n)  -f-  7. 

3.  19(3a- 66  +  9c)--3.       6.    12  a(3  6  -  3c  +  9)  h- 3. 

7.  24(166-8c  +  24d)-^8. 

8.  Divide  7  ct  •  14  6  •  21  c  by  7  in  three  different  ways. 

9.  Add  5  a,  -^ ,  --— ,  and  — ^ ,  using  Principles  V  and  L 

2         o  6 

10.  Prom  y  subtract  — '-^ ,  using  Principles  V  and  I. 

--     -n,         14 a,  10 a      1,       ,  Q>a 

11.  J^rom  - — ■  -\ subtract  — . 

2  5  3 

12.  Find  the  sum  of  — ,  — ,  — ,  1  x,  and  3  a;. 

8    '    5    '    4  ' 

1  o     T?-   J  ^v,  £  100  rs    90  rs        -,  2^  rs 

13.  Find  the  sum  of  ,  ,  and 

10     '      9  5 

14.  From  25  xy  subtract  — ^  • 

z 

-le  a;i^  8a6c  ,  IS  ah  ,  7  ahd  ,  ahe 

15.  Add 1 1 -| 

c  'S  d  e 

16  Add  1§_£*  I  5_^  I  "^  ^^' _i_  "^ 

6  h  X         m 

17  Add  5^_|_^8^  ,  32^  24^ 

28    12    4    3  ° 


20  INTRODUCTION"  TO   ALGEBRA 

ORDER  OF  INDICATED   OPERATIONS 

22.  Order  of  Operations.  In  a  succession  of  indicated  opera- 
tions the  final  result  depends  in  some  cases  upon  the  order  in 
which  the  operations  are  performed,  while  in  some  cases  it  does 
not.     This  is  shown  in  the  following  examples: 

(1)  6  +  3-8  would  give  9  •  8  =  72,  if  the  addition  were  performed 
first ;  and  would  give  6  +  24  =  30,  if  the  multiplication  were  performed 
first. 

(2)  24  -^  2  •  3  gives  12  •  3  =  36,  if  the  division  is  performed  first ;  and 
gives  24  -4-  6  =  4,  if  the  multiplication  is  performed  first. 

(3)  24  -^  6  -i-  2  equals  2  if  we  first  divide  24  by  6,  and  equals  8  if  we 
first  divide  6  by  2. 

(4)  12x6-4-3  =  24  no  matter  in  what  order  the  operations  are 
performed. 

(5)  18  +  10  —  8  =  20,  no  matter  in  what  order  the  operations  are 
performed. 

(6)  4  X  3  X  2  =  24,  no  matter  in  what  order  we  multiply. 

23.  To  prevent  mistakes,  and  to  make  usage  uniform,  the 
following  rules  have  been  adopted  r 

In  ayi  expression  involving  additions,  subtractions,  multiplica- 
tions, and  divisions,  when  no  symbols  of  aggregation  (§  11)  are 
involved, 

(1)  All  multiplications  are  ijerformed  Jirst,  and  these  may  be 
taken  in  any  order; 

(2)  All  divisions  are  2Jerformed  next,  and  these  are  taken  in  the 
order  in  which  they  occur  from  left  to  right ; 

(3)  Finally,  additions  and  subtractions  are  performed,  and 
these  may  be  taken  in  any  order. 

These  rules  are  illustrated  by  the  following  examples : 

(1)5  + 3 -4-8-2  =  5  + 12 -8-2  =  5+12-4  =  13; 

(2)  8-2-2-2x2+3=8-2-2-4+3  =  8-  1-4  +  3  =  6; 

(3)  8  -  2  -  4  +  28  -  2  -  2  =  16  -  4  +  28  -  4  =  4  +  7  =  11 ; 

(4)  16  +  24  -  3  -  4  -  8  -  3  -  2  =  16  +  24-12-24-2 
=  16  +  2  -  12  =6. 


ORDER   OF   INDICATED   OPERATIONS  21 

ORAL  EXERCISES 

1.  What  is  the  value  of  4- 3  +  2- 4-3- 5? 

2.  What  is  the  value  of  5- 4-6  +  2- 5? 

3.  What  is  the  value  of  2- 3- 4-}- 12 --4? 

4.  What  is  the  value  of  3  (4  +  2)  +  6  ? 

5.  What  is  the  value  of  2  (3  +  2  •  3)  -  4  (1  +  3)  ? 

If  a  =  2,  6  =  3,  c  =  4,  find  the  value  of  each  of  the  following 
Remember  that  a  fraction  is  an  indicated  division. 


6. 

2ah-c. 

11. 

9  a  +  3  c  -  10  6, 

7. 

he  —  A:  a 
a 

12. 

8a  -hc  +  5  6 

— • 

5 

8. 

3  ac  4-  2  6 

26 

13. 

4c-26  +  3a 
4a 

9. 

5a+25- 

3  c. 

14. 

2a  +  35  +  4c. 

10. 

10  c -10  6 

—  4  a. 

15. 

Qa  —  h  —  c. 

WRITTEN  EXERCISES 

If  a  =  1,  h  —  2,  c  =  3,  a;  =  4,  2/  =  5,  2;  =  6,  find  the  value  of 
each  of  the  following  expressions  : 

16.  (6a6--4)^3.  29.  2hz-{2x-c). 

17.  6  ah  H-  (4  -  3).  30.  2hz-2{x-  c). 

18.  {12  ex  ^  2  z)h.  31.  2hz-2x-c. 

19.  12  chx^  2  z.  32.  30  — (x— a). 

20.  12  ca;  ^  2  zh.  33.  30  —  ic  —  a. 

21.  {^xyz^2h)^2G.  34.  3  +  5  •  6  -  4  •  3. 

22.  Q>xyz-^{^h^2c),  35.  3 +  5(6 -4).  3. 

23.  5  ahc  —  3  x  -\-  2  yz.  36.  5(a  +  6  +  c—  x). 

24.  3  xyz  —  8  a  +  5  6.  37.  2  cicy  -i-  a6  +  5. 

25.  3  xyz  —  (8  a+  5  6.)  38.  4.  xy  ^  S  ah -\-  4. 

26.  3  axy  —  i  c  —  h.  39.  4  xy  ^  (4  a  ^  h). 

27.  3  axy  -  4  (c  -  h).  40.  2  aicy  -=-  (5  6  -^  2  a). 

28.  3rtx^  — 4(c  +  6).  41.  (2  axy -!- 5  6) -^  2  a. 


22  INTRODUCTION   TO   ALGEBRA 

24.  Importance  of  the  Principles.  The  five  principles  studied 
in  this  chapter,  together  with  others  which  will  be  introduced 
when  needed,  will  be  found  of  increasing  importance  as  we 
proceed.  Your  success  in  the  further  study  of  algebra  will 
depend  in  no  small  degree  upon  the  clearness  with  which  you 
understand  their  real  significance. 

The  most  effective  way  to  master  these  principles  is  by 
means  of  simple  numerical  examples  such  as  were  used  in 
introducing  each  one.  Make  a  list  of  these  principles  in  ab- 
breviated form  for  yourself  and  note  how  frequently  your 
own  errors  and  those  of  your  classmates  are  due  to  direct  viola- 
tions of  one  or  more  of  them. 

WRITTEN  EXERCISES 

If  a  =  5,  6  =  3,  c  =  2,  find  the  value  of  each  of  the  following : 

1.  ac  -}-  be  and  (a  +  h)  c.  5.    a(b  —  c)  and  ab  —  ac. 

2.  ac  -  be  and  (a  -b)c.  6.   —,-xb  and  ax  -• 

^  ^  c     c  c 

3.  abc,  a  x  (be)  and  b  X  (ac).  7. and  -  H 

^  c  c      c 

4.  a(b  +  c)  and  ab  -\-  ae.  8.    ^^— ^  and  -  —  -  • 

c  c      c 

9.    In  each  of  the  above  compare  the  results  obtained  in  the 

two  parts. 

4a-56  ,  6  +  9  ,,     12bc-\-Ab  ,  3ac-\-2b 

10.    ^ H  — 


o 


4 


86 -9c  ,  3a +  6 
11.    — ;^ f- 


12. 


3  6 

8a6-56c     2a6  +  3c 
6  6 


6ac  -f-  56c  .  a6  +  3c 
2a  6 

,,     12a-|-56  ,  6c-h46 


J.O. 

12         *         6 

16. 

10a6-3c     56c -2a 
26                  a 

17. 

45  +  6a  -  3c 
6 

18. 

12c  +  4ac  —  4c 

8 

19. 

(6a  +  i)c)  -7-  2a 

REVIP]W   QUESTIONS  23 

REVIEW   QUESTIONS 

1.  How  would  3  •  5  and  7  •  5  be  added  in  arithmetic  ?  Why 
cannot  3  n  and  7  n  be  added  in  the  same  manner  ?  State  the 
principle  by  which  3  n  and  7  n  are  added.  Test  the  identity 
3  n  +  7  7i  =  10  71  by  substituting  any  convenient  value  for  n. 

2.  What  kind  of  numbers  may  be  added  by  Principle  I  ? 
Have  the  numbers  ac  and  be  a  common  factor  ?  What  is 
it  ?  What  is  the  coefficient  of  this  common  factor  in  each  ? 
What  is  the  sum  of  these  coefficients  ?  Is  the  equality 
ac  -{-  be  =  (a  -]-  b)  c  true  no  matter  what  numbers  are  repre- 
sented by  a,  b,  and  c?  When  this  can  be  said  of  an  equality, 
what  is  it  called  ? 

3.  How  is  5  •  9  subtracted  from  11  •  9  in  arithmetic  ?  In 
what  different  manner  may  this  operation  be  performed  ?  Why 
is  it  sometimes  necessary  to  perform  subtraction  in  the  second 
way  ?  In  the  identity  31  x—  12  x  =  19  x,  what  number  is  rep- 
resented by  X  ?  Test  the  equality  by  substituting  any  conven- 
ient number  for  x.     Is  this  equality  true  for  every  value  of  a:  ? 

Principle  I  may  be  conveniently  abbreviated  as  follows : 

ac  -j-  be  =  (a-i-b)c, 
ac  —  be  =  {a  —  b)o. 

4.  How  is  11  +  3  multiplied  by  4  in  arithmetic  ?  In  what 
different  way  may  this  operation  be  performed  ?  Why  is  it 
sometimes  necessary  to  multiply  in  the  second  way  ?  State  in 
full  the  principle  by  which  a  +  8  is  multiplied  by  7. 

Principle  II  may  be  abbreviated  thus : 

c{a  -i-  b)  =  ca  -\-  cb, 
c(a  —  b)  =  ca  —  cb. 

Notice  that  the  identities  in  Principle  II  are  the  same  as 
those  in  Principle  I  read  in  reverse  order.  Thus,  Principle  II 
may  be  called  the  reverse  or  converse  of  Principle  I. 


24  INTRODUCTION  TO  ALGEBRA 

5.  How  is  12-1-18  divided  by  6  in  arithmetic?  In  what 
different  way  may  this  division  be  performed  ?  Why  is  it 
sometimes  necessary  to  perform  division  in  the  second  way  ? 
State  in  full  the  principle  used  in  performing  the  operation 
(6  ic  +  9  ?/)  ^  3.     How  do  you  divide  (6x-9y)by  S? 

Principle  III  may  be  abbreviated  thus : 

a  -\-  b  _a      b  a  —  b      a      b 


k         k     k  k         k     k 

6.  How  is  the  product  2  •  3  •  5  multiplied  by  4  in  arithmetic  ? 
In  what  different  way  may  this  multiplication  be  performed  ? 
Why  is  it  ever  performed  in  the  second  way  ?  What  are 
the  factors  of  ah  ?  How  is  the  product  of  two  numbers  multi- 
plied by  another  number?  Should- 6o^7i  factors  be  multiplied 
by  the  number,  or  only  one  ?  Is  it  permissible  to  multiply 
either  one  we  choose  ? 

Principle  IV  is  abbreviated  thus  : 

k  X  {ab)  =  (ka)  xb=  a  x  (kb). 

7.  Divide  2  •  4  •  6  •  20  by  2  without  first  performing  the 
multiplication  indicated  in  2  •  4  •  6  •  20.  Do  this  in  several 
ways  and  show  that  all  the  quotients  obtained  are  equal.  State 
in  full  the  principle  used. 

Principle  V  is  abbreviated  thus : 

(ab)-^k  =  ^xb  =  ax~' 
k  k 

8.  Contrast  Principles  II  and  IV ;  also  III  and  V. 

9.  Why  are  our  numerals  called  Arabic  numerals  ?     When 
and  how  were  they  brought  into  Europe? 

10.  What  is  the  most  important  point  in  which  the  Arabic 
system  differs  from  the  Roman  system  ? 

11.  What  is  meant  by  the  distributive  law  of  multiplication 
and  the  distributive  law  of  division  ? 


CHAPTER   TI 

EQUATIONS  AND  PROBLEMS 

The  principles  developed  in  the  last  chapter  will  now  be 
used  in  the  solution  of  equations  and  problems. 

SOLUTION  OF  EQUATIONS 

25.  Unknowns.  A  letter  whose  value  we  seek  by  means  of 
an  equation  is  called  an  unknown  in  that  equation. 

Satisfying  an  Equation.  A  number  is  said  to  satisfy  an  equa- 
tion if,  when  substituted  for  the  unknown,  it  reduces  the  equa- 
tion to  an  identity. 

Thus,  3  X  =  18  is  said  to  be  satisfied  by  a;  =  6,  because  this  value  of  x 
reduces  the  equation  to  the  identity  18  =  18. 

Equivalent  Equations.  If  two  equations  which  are  not  identi- 
ties are  satisfied  by  the  same  numbers,  they  are  called  equiva- 
lent equations. 

Thus,  3  X  =  18  and  6  x  =  36  are  equivalent  equations  because  they  are 
not  identities  and  are  both  satisfied  by  x  =  6. 

Also,  7i  +  2  =  5  and  2  u  +  4  =  10  are  equivalent  equations,  because 
they  are  both  satisfied  by  7i  =  3  and  by  no  other  value  of  n. 

ORAL  EXERCISES 

1.  Is  a:  -f-  4  =  9  satisfied  by  a;  =  4  ?  by  ic  =  5  ?  by  x  =  6  ? 

2.  Is  2.-^  +  9  =  .T-f  12  satisfied  by  it'  =  2?   by.T  =  3? 

3.  Is  -4- 3  =  6  satisfied  by  a;  =  4?  by  .i;  =  8  ?  by  a;=12? 

•± 

4.  Is  3(.T  +  2)  =  3a;  +  6  satisfied  by  x  =  l?  by  x=2?  by 
a;  =  3?  bya;=:4? 

25 


26  EQUATIONS   AND   PROBLEMS 

26.  To  solve  an  equation  is  to  find  a  value  of  the  unknown 
which  will  satisfy  the  equation. 

A  value  of  the  unknown  which  satisfies  an  equation  is  called 
a  root  or  solution  of  the  equation. 

The  following  examples  illustrate  methods  used  in  solving 
equations. 

Example  1.     Solve  the  equation  a;  —  5  =  9.  (1) 

Solution.  This  equation  states  that  9  is  5  less  than  the  number  x. 
That  is,  if  9  be  increased  by  5,  the  result  is  x. 

Hence,  x  =  14  is  the  solution  of  equation  (1). 

This  result  may  he  obtained  by  adding  5  to  each  member  of  equation  (1), 
thus  x  +  5-6  =  9  +  5, 

or  X  =  14.  (2) 

Example  2.     Solve  the  equation  x-\-l  =12.  (1) 

Solution.  This  equation  states  that  12  is  7  more  than  the  number  x. 
That  is,  if  12  be  diminished  by  7,  the  result  is  x. 

Hence,  x  =  5  is  the  solution  of  equation  (1). 

This  result  may  he  obtained  by  subtracting  1  from  both  members  of  (1), 
thus  x  +  7-7  =  12-7, 

or  X  =  b.  (2) 

Example  3.    Solve  the  equation  ijc  =  7.  (1) 

Solution.  This  equation  states  that  one  third  of  the  number  x  is  7. 
That  is,  X  is  3  times  7,  or  21. 

Hence  x  =  21  is  the  solution  of  equation  (1). 

This  result  may  be  obtained  by  multiplying  both  members  of  (1)  by  3, 
thus  3  X  ^x  =  3  .  7, 

or  X  =  21.  (2> 

Example  4.    Solve  the  equation  5  ic  =  30.  (1) 

Solution.  This  equation  states  that  5  times  the  number  x  is  30.  That 
is,  X  is  one  fifth  of  30,  or  6. 

Hence,  x  =  G  is  the  solution  of  the  equation. 

TTiis  result  may  be  obtained  by  dividing  both  members  of  eqxiation  (1) 
by  5,  thus  b_x  _  30 

5    ~  s' 
or  X  =  6.  (2) 


SOLUTION  OF   EQUATIONS  27 


ORAL  EXERCISES 


Solve  the  following  equations  and  explain  each  step  involved, 
as  in  the  foregoing  illustrative  examples. 


1. 

2a;  =16. 

2. 

11  a;  =  33, 

3. 

iy  =  6. 

4. 

ix  =  7. 

5. 

171  =  20. 

6. 

6w  =  A2. 

7. 

.T  +  4  =  8.    . 

13. 

w-{-2  =  14. 

8. 

x-\-9  =  16. 

14. 

w-2  =  14. 

9. 

.T  -  2  =  8. 

15. 

iw;  =  20. 

10. 

a;-3  =  10. 

16. 

7?(;  =  28. 

11. 

a.'  +  8=16. 

17. 

3n-l  =  5. 

12. 

a;  -  3  =  12. 

18. 

3  ?i  +  1  =  10. 

27.  Deriving  Equivalent  Equations.  The  above  examples  il- 
lustrate four  ways  of  operating  upon  an  equation,  so  as  to  pro- 
duce in  each  case  a  new  equation  ivhich  is  equivalent  to  the  origi- 
nal equation. 

Each  of  these  operations  changes  the  value  of  both  members, 
but  changes  them  both  in  the  same  way. 

The  object  of  each  step  is  to  obtain  an  equation  whose 
solution  is  more  apparent  than  that  of  the  preceding  equation. 

In  the  example  below,  equations  (2)  and  (3)  show  two  ways 
of  changing  the  form,  but  7iot  the  value,  of  one  member  alone, 
thus  producing  a  new  equation  which  is  equivalent  to  the 
original  equation. 

Example  5.    Solve  the  equation 

zo  +  2  (id  +  5)  =  58.  (1) 

By  Principle  II,  to  +  2  to  +  10  =  68.  (2) 

By  Principle  I,  3  w;  +  10  =  58.  (3) 

Subtracting  10  from  both  members,         3  to  =  48.  (4) 

.Dividing  both  members  by  3,  lo  =  IG.  (5) 

Check.      Putting  to  =  16   in    (1),     16  +  2(16  +  5)  =  16  +  2  .  21  =  58. 

The  operations  involved  in  passing  from  (1)  to  (2)  and  (2) 
to  (3)  are  called  form  changes. 

All  the  operations  involved  in  Principles  I  to  V  are  f>rni 
changes  of  this  character.  See  the  list  at  the  end  of  Chapter  I. 
Other  form  changes  will  be  considered  as  need  arises. 


28 


EQUATIONS   AND   PROBLEMS 


28.   Illustrating  the  Operations  on  an  Equation.     The  members 
of  an  equation  may  be  likened  to  the  scale-pans  of  a  common 

balance  in  which  are  placed 
objects  of  uniform  weight, 
say  tenpenny  nails.  The 
scales  balance  only  when  the 
weights  are  the  same  in  both 
pans  ;  that  is,  when  the  num- 
ber of  nails  is  the  same. 


ORAL  EXERCISES 

1.  If  there  are  ten  nails  on  each  side,  do  the  scales  balance  ? 

2.  If  now  six  more  nails  are  added  to  one  side,  what  must 
be  done  to  the  other  side  to  keep  the  scales  balanced  ?  If 
three  nails  are  subtracted  from  one  side  what  must  be  done  to 
the  other  side  to  keep  the  scales  balanced  ? 

3.  If  the  number  of  nails  on  one  side  is  doubled,  what  must 
be  done  to  the  other  side  to  keep  the  balance  ?  If  the  number 
is  divided  by  four  ? 

4.  If  the  scales  are  balanced,  and  if  the  nails  in  either  pan 
are  rearranged  in  any  way,  will  the  scales  continue  to  balance  ? 

Preserving  the  Balance.  The  above  examples  show  that  if  the 
scales  are  in  balance,  they  will  remain  so  under  two  kinds  of 
changes  in  the  weights  : 

(a)  When  the  number  of  nails  in  the  two  pans  is  increased 
or  diminished  by  the  same  amount ;  corresponding  to  like 
changes  in  value  of  both  members  of  an  equation. 

(6)  When  the  number  in  each  pan  is  left  unaltered  but  the 
nails  are  rearranged  in  groups  or  piles  in  any  manner;  corre- 
sponding to  form  changes  in  either  member  of  an  equation. 

The  equation,  then,  is  like  a  balance,  and  its  meinbers  are  to  be 
operated  upon  onh/  in  such  ivays  as  to  ^weserve  the  balance. 


SOLUTION   OF   EQUATIONS  29 

WRITTEN   EXERCISES 

Solve  the  following  equations  as  in  example  5,  §  27 : 

1.  a;  +  4a;  =  15.  7.  3(4r-2)  =  18. 

2.  12a;  +  3a;  =  30.  8.  y -{-2(y +  2)=16. 

3.  7a^-3a;  =  12.  9.  r  +  3(r  +  4)=24. 

4.  a;  +  3(x-l)  =  17.  10.  G?  +  2(Z-3)  =  34. 

5.  4(3 .T  —  a;)  =  8.  11.  -i-w  +  4w  =  26. 

6.  5(2a;  +  l)=15.  12.  ^iv-^2w-\-3  =  13. 

The  foregoing  examples  illustrate 

Principle  VI 

29.  Rule.  Ajz  equation  may  be  changed  into  an  equiva- 
lent equation  hy  means  of  any  of  the  following  operations  : 

(1)  Adding  the  same  number  to  both  inembers ; 

(2)  Subtracting  the  same  number  from  both  members ; 

(3)  Multiplying  both  members  by  any  known  number 
not  zero; 

(4)  Dividing  both  inembers  by  any  known  nuinber  not 
zero ; 

(5)  Changing  the  form  of  either  member  in  any  way 
which  leaves  its  value  unaltered. 

The  operations  under  Principle  VI  are  hereafter  referred  to 
in  detail  by  rieans  of  the  initial  letters,  A  for  addition,  S  for 
subtraction,  3/ for  midtiplication,  D  for  division,  and  F  fov  form 
changes  which  leave  the  value  of  a  member  unaltered. 

Note.  — The  statements  (1)  to  (4)  in  Principle  VI  include  the  following 
so-called  Axioms  or  self-evident  truths  : 

(1)  If  equals  are  added  to  equals,  the  sums  are  equal. 

(2)  If  equals  are  subtracted  from  equals,  the  remainders  are  equal. 

(3)  If  equals  are  multiplied  by  equals,  the  products  are  equal. 

(4)  If  equals  are  divided  by  equals,  the  quotients  are  equal. 
Principle  VI,  however,  includes  more  than  these  axioms,  since  it  states 

that  the  new  equation  obtained  each  time  is  equivalent  to  the  original 
equation. 


30  EQUATIONS    AND    PROBLEMS 

DIRECTIONS  FOR  WRITTEN   WORK 

30.   In  solving   an   equation   the    successive   steps   may  be 
written  as  in  the  following  examples  : 

1.  25(7i  +  l)  +  6(4n-3)=50  +  31?i  +  2(3-n)'-9.  (1) 

By  F^  using  Principle  II,  we  obtain  from  (1) 

25  n  +  25  +  24  n  -  18  =  50  +  31 71  +  6  -  2  71  -  9.  (2) 

By  F^  using  Principle  I,  we  obtain  from  (2) 

49n  +  7  =  29n+47.  (3) 

Subtracting  7  and  29  ri  from  each  member  of   (3)  and  using  Princi- 
ple I,  we  have  20  ri  =  40.  (4) 
Dividing  each  member  of  (4)  by  20, 

n  =  2.  (5) 

Check.     Substitute  7i  =  2  in  equation  (1). 

For  convenience  this  work  can  be  abbreviated  as  follows  : 

25(n  +  1)  +  G(4  7i  -  3)  =  50  +  31  n  +  2(3  -  n)  -  9.  (1) 

By  F,  II,      25  n  +  25  +  24  71  -  18  :=  50  +  31 7i  +  6  -  2  ?i  -  9.  (2) 

By  F,  I,  49  71  +  7  ==  29  n  +  47.  (3) 

By  5-17,29  71,  20  71  =  40.  (4) 

By  Z)  1 20,  71  =  2.  (5) 

>S'  I  7,  29  71  means  that  7  and  29  n  were  subtracted  from  both  members  of 

the  preceding  equation.     D  I  20  means  that  the  members  of  the  preceding 

equation  were  divided  by  20. 

Similarly,  in  case  we  wish  to  indicate  that  6  is  to  be  added  to  eacli 

member  of  an  equation,  we  should  write  A  I  6,  and  if  each  member  is  to 

be  multiplied  by  8,  we  should  write  M\  8. 

2.  17  7i  +  4(2  +  7i)-6  =  5(4  +  7i)-5-f  371.  •  (1) 

By  i^,  II,         17  71  +  8+471-6  =  20  +  571-5  +  371.  (2) 

By  i?',  I,  21 71  +  2  =  15  +  8  n.  (3) 

By  /S- 1 2,  8  71,  2171-871  =  15-2.  (4) 

By  i?',  I,  13  71  =  13.  (5) 

By  D  1 13,  71  =  1.  (0) 

Check.     Substitute  7i  =  1  in  equation  (1). 

3.   l+l  +  x  =  n.  (1) 

By  itf  1 6,  2x  +  3x  +  Cx  =  G6.  (2) 

By  F,  I,  11  x  =  66.  (3) 

Byl>lll,  x  =  6.  (4) 


SOLUTION   OF   EQUATIONS  31 

31.  Transposing  Terms.  By  use  of  Principle  VI,  a  term  may 
be  transposed  from  one  member  of  an  equation  to  the  other, 
provided  its  sign  is  changed. 

E.g.  in  deriving  equation  (4)  from  (3)  in  Example  2,  page  .30,  8  n 
is  subtracted  from  both  sides  by  mentally  dropping  it  on  the  right  and 
indicating  its  subtraction  on  the  left.  Likewise  when  2  is  subtracted 
from  both  sides  it  disappears  on  the  left  and  appears  on  the  right  with 
the  opposite  sign.  Each  of  these  indicated  subtractions  might  have  been 
performed  mentally.,  thus  writing  equation  (5)  directly  from  (3). 

After  a  little  practice  this  shorter  process  of  transposing  terms, 
changing  the  signs,  and  combining  similar  terms  mentally  should 
always  be  used 

WRITTEN  EXERCISES 

Solve  the  following  equations,  putting  the  work  in  a  form 
similar  to  that  on  page  30.     Check  the  first  ten. 

1.  13x-^4:0-x  =  SS.  8.  42  x'  + 56  =  20a; +  122. 

2.  3a;  +  15  +  2.^■=:18  +  4.^•.  9.  12m -}-3 -3m  =  38  +  2??i. 

3.  5x-\-3  -  x  =  x-j-  18.  10.  15  m  +  3  —  2  m  =  3  «i  +  53. 

4.  13?/  + 12  +  5// =  32 +  8?/.  11.  rt +  7  +  3a  =  2a  +  45. 

5.  4m  +  6?7i  +  4  =  9//t +  0.  12.  5  6  +  30  +  66  =  36  +  150. 

6.  7m  +  18  +  3/«  =  2m  +  50.  13.  3c  +  18  +  14c  =  6  c-^  51. 

7.  v  +  42  +  45//  =  76  +  12.?/.  14.  17  ;r  +  4 +  3a;  =  7a;  +  30. 

15.  3  //  +  4  +  2  ?/  +  6  =  //  +  7  +  7/  +  3  +  30. 

16.  5  .1-  +  3  +  2  it'  +  3  =  2  a;  +  5  +  3  a;  +  3  +  x. 

17.  2  .y  +  4  a;  +  9  —  .t-  +  6  =  20  +  2  .y  +  5  +  x. 

18.  18  +  6  m  +  30  +  4  m  =  4  m  +  8+12  +  3  m  +  3  -  m  +  29. 

19.  6a;  +  8  +  .r  +  4  +  5  a-  =  7  x  +  32  -  x  -  20. 

20.  32  a;  +  4  +  7  .V  =  66  +  3  x  +  5  x. 

21.  7(w  +  6)+ 10  771  =  42  +  5 /;«,  + 24. 

22.  6a;  +  4(4a;  +  2)+3(2.v+7)=85. 

23.  8  +  7(6  +  6  /0+  2  ?i  =  2(4  n  +  5)+  18  »  +  49. 

24.  5(9a;  +  3)  +  4(3.i;  +  2)  =  18a;  +  36. 


32  EQUATIONS  AND  PROBLEMS 

32.  An  equation  may  be  translated  into  a  problem.  For  exam- 
ple, the  equation  21  a:  +  2  =  8a; -f  15  may  be  interpreted  as 
follows :  Find  a  number  such  that  21  times  the  number  plus  2  is 
15  greater  than  8  times  the  number. 

ORAL  EXERCISES 

Translate  each  of  the  following  into  a  problem  : 

1.  2a; +  5  =  21.  5.    3 ( a; -|- 1)  =  18. 

2.  3a; -7  =  19.  6.    2a;  +  3  +  a;  =  .^•  +7. 

3.  6  -f  5  a;  =  16.  7.    5  a;  —  3  —  a;  =  .?;  +  9. 

4.  4a;-3  =  2a;  +  9.  8.    8  +  3a;+ 2a;  =4a;-f  19. 

HISTORICAL  NOTE 

Origin  ot  the  Name  Algebra.  The  Arabs  brought  their  first  algebra  into 
Europe  in  the  first  half  of  the  ninth  century.  It  bore  the  name  Al-jebr- 
w^l-muqabala.  The  word  al-gehr,  from  which  the  word  algebra  is  de- 
rived, means  "  transposition  "  and  refers  to  transposing  terms  in  solving 
equations.  The  word  loal-muqdhala  means  the  process  of  simplification 
or  form  changes.  Thus  it  appears  that  the  Arabs  regarded  the  solution  of 
equations  as  the  main  business  of  algebra. 

SOLUTION   OF  PROBLEMS 

33.  One  great  object  in  the  study  of  algebra  is  to  simplify 
the  solution  of  problems.  This  is  done  by  using  letters  to  repre- 
sent numbers,  by  stating  problems  in  the  form  of  equations, 
and  by  the  systematic  solutions  of  the  equations. 

Illustrative  Problem.  1.  The  shortest  railway  route  from 
Chicago  to  New  York  is  912  miles.  How  long  does  it  take  a 
train  averaging  38  miles  an  hour  to  make  the  journey  ? 

Solution.  Let  t  be  the  number  of  hours  required.  Then  38  f  is  the 
distance  traveled.     But  912  miles  is  the  given  distance  traveled. 

Hence  38^  =  012.  (1) 

By  i)  1 38  «=24.  (2) 

Check.     The  distance  traveled  in  24  houi*s  at  38  miles  per  hour  is 

38  X  24  =  912. 
Hence  <  =  24  satisfies  the  conditions  of  the  problem. 


SOLUTION   OF  PROBLEMS  33 

Illustrative  Problem.  2.  For  liow  many  years  must  S  850 
be  invested  at  5  %  simple  interest  in  order  to  yield  $  255  ? 

Solution.     Erom  arithmetic,  we  have 

principal  x  rate  x  time  —  interest, 
or  prt  =  i. 

Hence,  from  the  conditions  of  the  problem, 

850  X  .05  xt  =  255,  (1) 

or  42.5 1  =  255. 

By  D  I  42.5,  t  =  6.  (2> 

Check.     The  interest  on  ^  850  for  6  years  at  5  %  is 
6  X  .05  X  850  =  255. 

Hence  t  =  6  satisfies  the  conditions  of  the  problem. 

Note.  —  Substituting  t  =  Q  in.  equation  (1)  would  not  fully  check  the- 
solution,  since  that  equation  might  be  incorrect.  It  is  necessary  to  see 
that  the  solution  satisfies  the  problem  itself. 

Illustrative  Problem.  3.  A  boy,  an  apprentice,  and  a  master 
workman  have  the  understanding  that  the  apprentice  shall 
receive  twice  as  much  as  the  boy  and  the  master  workman  five 
times  as  much  as  the  boy.  How  much  does  each  get,  if  the 
total  amount  received  for  a  piece  of  work  is  $  104  ? 

Solution.     Let  n  represent  the  number  of  dollars  received  by  the  boy. 
Then,     2  n  is  the  number  of  dollars  received  by  the  apprentice, 
and  5  n  is  the  number  of  dollars  received  by  the  master  workman. 

Hence,  n  +  2  n  +  5  w  represents  the  total  amount  received. 
Therefore,  n  +  2n  +  6n=  104.  (1> 

By  Principle  I,  Sn  =  104.  (2) 

By  D  I  8,  n  =  13.  (3> 

Hence,  the  amount  received  by  the  boy  is  13  dollars.  The  appren- 
tice receives  2  n  dollars  or  $  26,  and  the  master  workman  receives  5  n 
dollars  or  $65. 

Check.  By  the  conditions  of  the  problem  the  sum  of  the  amounts 
obtained  should  be  §  104  ;  the  apprentice  should  receive  twice  as  much 
as  the  boy  and  the  master  workman  five  times  as  much  as  the  boy. 
That  is,  we  should  have 

13  +  26  +  65  =-.  104,     26  =  2  .  13     and  66  =  5  •  13. 


34  EQUATIONS   AND   PROBLEMS 

Rules  for  Solving  Problems.  From  the  preceding  illustrations 
we  see  how  much  more  simply  problems  may  be  solved  by 
algebra  than  by  arithmetic. 

The  following  rules  will  help  in  solving  problems : 

(1)  Represent  the  wiknown  quantity  by  a  letter. 

(2)  Trayislate  the  words  of  the2)roblems  into  an  equation  involv- 
ing this  letter. 

(3)  Solve  this  equation  by  use  of  Principle  VI. 

(4)  Verify  the  solution  by  shoioing  that  the  requirements  of  the 
p)roblem  are  fuljilled. 

ORAL  PROBLEMS 

1.  Three  times  a  certain  number  is  24.  What  is  the 
number  ? 

If  X  is  the  number,  then  3  a;  =  24. 

2.  If  ^  of  a  certain  number  is  5,  find  the  number.  First 
state  the  equation. 

In  each  of  the  following,  state  the  equation  and  then  give 
the  required  number. 

3.  If  6  is  added  to  a  certain  number,  the  sum  is  22.  Find 
the  number. 

4.  If  8  is  subtracted  from  a  certain  number,  the  remainder 
is  10.     Find  the  number. 

5.  Twice  a  certain  number  added  to  the  number  itself 
gives  12.     Find  the  number. 

6.  Three  times  a  number  less  the  number  itself  equals  18. 
Find  the  number. 

7.  Twice  a  number  plus  three  times  the  number  equals  25. 
Find  the  number. 

8.  Six  times  a  number  less  4  times  the  number  equals  8. 
Find  the  number. 

9.  If  3x  plus  4  a;  equals  42,  find  the  value  of  x. 


SOLUTION   OF   PROBLEMS  35 

10.  If  2  a  is  subtracted  from  9  a,  the  remainder  is  21.  Find 
the  value  of  a. 

11.  If  7  X  is  subtracted  from  12  x,  the  remainder  is  15.  Find 
the  value  of  x. 

12.  If  4  2/  is  added  to  6  y,  the  sum  is  100.    Find  the  value  of  y. 

13.  If  Zx,  4x,  and  %x  are  added,  the  sum  is  30.  Find  the 
value  of  x. 

14.  Twice  a  number  plus  three  times  the  number  plus  four 
times  the  number  is  36.     Find  the  number. 

15.  If  n  is  a  number,  how  do  you  represent  10  times  that 
number  ? 

16.  If  n  is  a  number,  how  do  you  represent  that  number 
plus  3  times  itself  ? 

17.  If  71  is  a  number,  how  do  you  represent  5  times  that 
number  plus  3  times  the  number  plus  8  times  the  number  ? 

WRITTEN  PROBLEMS 

Solve  the  following  problems  by  means  of  equations,  and 
check  each  result  by  testing  whether  it  satisfies  the  conditions 
of  the  problem. 

1.  Five  times  a  certain  number  equals  80.     What  is  the 
number  ? 

2.  Twelve  times  a  number  equals  132.    What  is  the  number  ? 

3.  A  tank  holds  750  gallons.     How  long  will  it  take  a  pipe 
discharging  15  gallons  per  minute  to  fill  the  tank  ? 

4.  The  cost  of  paving  a  block  on  a  certain  street  was  $  7  per 
front  foot.    How  long  was  the  block,  if  the  total  cost  was  $  4620  ? 

5.  A  city  lot  sold  for  $  7500.     What  wag  the  frontage,  if  the 
selling  price  was  $  225  per  front  foot  ? 

6.  An   encyclopedia   contains   18,000   pages.     How   many 
volumes  are  there,  if  they  average  750  pages  to  the  volume  ? 


36  i:quati()ns  and  problems 

7.  For  how  many  years  must  $3500  be  invested  at  6  % 
simple  interest  to  yield  $  2205  ? 

8.  At  what  rate  must   $2500  be  invested  for  3  years  in 
order  to  yield  $412.50? 

Suggestion.    By  the  conditions  of  the  problem,  2500  x  r  x  3  =  412.50. 

9.  At  what  rate  must  $  6800  be  invested  for  7  years  in  order 
to  yield  $2380? 

10.  How  many  dollars  must  be  invested  for  5  years  at  41% 
simple  interest  to  yield  $  351  ? 

Suggestion.    By  the  conditions  of  the  problem,  p  x  .04|  x  5  =  351. 

11.  How  many  dollars  must  be  invested  for  6  years  at  4|% 
simple  interest  to  yield  $2422.50? 

12.  A  cut  in  an  embankment  is  500  yards  long  and  4  yards 
deep.     How  wide  is  it  if  18,760  cubic  yards  are  removed  ? 

Suggestion.     By  the  conditions  of  the  problem,  500  x  to  x  4  =  18,760. 

13.  How  deep  is  a  rectangular  cistern  which  holds  500  cubic 
feet  of  water,  if  it  is  6  feet  wide  and  8  feet  long  ? 

14.  How  long  is  a  box  containing  2240  cubic  inches,  if  its 
width  is  14  inches  and  its  depth  10  inches  ? 

15.  The  greater  of  two  numbers  is  5  times  the  less,  and 
their  sum  is  180.     What  are  the  numbers  ? 

16.  A  number  increased  by  twice  itself,  4  times  itself,  and 
6  times  itself,  becomes  429.     What  is  the  number  ? 

17.  A  father  is  3  times  as  old  as  his  son,  and  the  sum  of 
their  ages  is  48  years.     How  old  is  each  ? 

18.  In  a  company  there  are  39  persons.  The  number  of 
children  is  twice  the  number  of  grown  people.  How  many 
are  there  of  each  ? 


SOLUTION   OF   PROBLEMS  37 

34.  Further  Hints  for  Translating  Problems  into  Equations. 
Skill  in  solving  problems  depends  upon  attention  to  the 
following  points : 

(1)  Mead  and  understand  clearly  the  statement  of  the  prol> 
lem,  as  it  is  given  in  words. 

(2)  Represent  the  unknown  number  by  some  letter,  say  the 
initial  letter  of  a  word,  which  will  keep  its  meaning  in  mind. 
If  there  are  more  unknown  numbers  than  one,  try  to  express 
the  others  in  terms  of  the  letter  first  selected. 

(3)  Form  two  algebraic  expressions  which,  according  to  the 
conditions  of  the  problem,  represent  the  same  number,  and  set 
them  equal  to  each  other,  thus  forming  an  equation. 

ORAL  EXERCISES 

1.  If  n  is  a  number,  represent  in  symbols  a  number  7 
greater  than  n ;  5  less  than  n ;  8  times  as  great  as  n ;  one  third 
as  great  as  n. 

2.  Write  in  symbols  n  increased  by  A: ;  n  decreased  by  k ; 
n  multiplied  by  A;;  n  divided  by  k. 

3.  If  the  sum  of  two  numbers  is  10  and  one  of  them  is  Xy 
what  is  the  other  number  ? 

4.  If  two  numbers  differ  by  6  and  the  smaller  is  x,  what  is 
the  other  number  ? 

5.  If  two  numbers  differ  by  6  and  the  greater  is  x,  what  is 
the  other  number  ? 

6.  If  A  has  m  dollars,  and  B  has  15  dollars  more  than  A, 
how  do  you  represent  B's  money  ?  If  C's  money  is  twice  B's, 
how  do  you  represent  C's  money  ? 

7.  A  father  is  34  years  older  than  his  son.  If  x  repre- 
sents the  age  of  the  son,  how  do  you  represent  the  father's 
age  ?     How  do  you  represent  their  ages  5  years  ago  ? 


38  EQUATIONS   AND   PROBLEMS 

8.  If  n  is  an  integer,  how  do  you  represent  the  next  higher 
integer  ?  The  second  higher  ?  The  next  lower  ?  The  second 
lower  ? 

Note.  —  The  whole  numbers  or  integers  are  obtained  by  ordinary 
counting,  beginning  with  one.  If  any  integer  such  as  12  is  given,  the  next 
higher  integers  may  be  represented  by  12+1,  12+2,  12+3,  and  so  on. 
The  next  lower  integers  may  be  represented  by  12  —  1,  12  —  2,  12  —  3, 
and  so  on. 

9.  What  is  the  value  of  2  n,  for  n  ^  1,  2,  3,  4,  5,  6,  etc.  ? 
If  n  is  any  integer,  the  number  represented  by  2  n  is  called 
an  even  integer  sirice  it  contains  the  factor  2. 

10.  If  2  w  is  any  even  integer,  represent  the  next  higher 
even  integer. 

11.  Represent  each  of  four  consecutive  even  integers,  the 
smallest  of  which  is  2  n. 

12.  What  is  the  value  of  2n-\-l,  for  n  =  1,  2,  3,  4,  5,  etc.  ? 
If  n  is  any  integer,  the  number  represented  by  2  n  -\-l  is 
called  an  odd  integer  since  it  does  not  contain  the  factor  2. 

13.  If  2  ?i  +  1  is  any  odd  integer,  represent  the  next  higher 
odd  integer. 

14.  Represent  four  consecutive  odd  integers,  the  smallest  of 
which  is  2  ?i  -f- 1- 

15.  If  X  is  a  number,  express  in  terms  of  x  a  number  5  less 
than  3  times  x\  also  a  number  5  times  the  remainder  when  3 
is  subtracted  from  x. 

16.  If  w  and  I  are  the  width  and  length  respectively  of  a 
rectangle,  how  do  you  represent  its  perimeter  ?  (The  perimeter 
of  a  rectangle  means  the  sum  of  the  lengths  of  its  four  sides.) 

?"+5  17.    The  length  of  a  rectangle  is  3 

feet  greater  than  its  width.  If  w  is 
the  width,  how  do  you  represent  its 
length  ?    its  perimeter  ? 


\0 


p=2w+2(wi-3) 


u 


p  =  2I  +  2(l-10) 


SOLUTION   OF   PROBLEMS  39 

18.  Express  the  perimeter  of  a  rec- 
tangle in  terms  of  its  length  I  if 
its  width  is  10  inches  less  than  the 
length. 

19.  Express   the  perimeter  of  a  rectangle  iu  terms  of  its 
length  I,  if  the  length  is  6  inches  greater  than  the  width. 

WRITTEN   PROBLEMS 

Check  each  solution  by  finding  whether  the  result  satisfies 
the  conditions  stated  in  the  problem  : 

1.  Four  times  a  certain  number  plus  3  times  the  number 
minus  5  times  the  number  equals  48.     What  is  the  number  ? 

2.  One  number  is  4  times  another,  and  their  difference  is  9. 
What  are  the  numbers  ? 

3.  Find  a  number  such  that  when  4  times  the  number  is 
subtracted  from  12  times  the  number,  the  remainder  is  496. 

4.  Thirty-nine  times  a  certain  number,  plus  19  times  the 
number,  minus  56  times  the  number,  plus  22  times  the  num- 
ber, equals  12.     Find  the  number. 

5.  There  are  three  numbers  whose  sum  is  80.  The  second 
is  3  times  the  first,  and  the  third  twice  the  second.  What  are 
the  numbers  ? 

6.  There  are  three  numbers  such  that  the  second  is  11  times 
the  first  and  the  third  is  20  times  the  first.  The  difference 
between  the  second  and  third  is  36.     Find  the  numbers. 

7.  There  are  three  numbers  such  that  the  second  is  8  times 
the  first  and  the  third  is  3  times  the  second.  The  third  num- 
ber less  the  second  equals  48.     Find  the  numbers. 

8.  The  number  of  representatives  and  senators  together  in 
the  United  States  Congress  is  531.  The  number  of  represent- 
atives is  51  more  than  4  times  the  number  of  senators.  Find 
the  number  of  each. 


40  EQUATIONS   AND   PROBLEMS 

9.  The  area  of  Illinois  is  6750  square  miles  more  than  10 
times  that  of  Connecticut.  The  sum  of  their  areas  is  61,640 
square  miles.     Find  the  area  of  each  state. 

10.  Find  three  consecutive  integers  whose  sum  is  144. 

11.  Find  four  consecutive  integers  such  that  the  last  plus 
twice  the  first  equals  48. 

12.  Find  three  consecutive  even  integers  whose  sum  is  54, 

13.  Find  three  consecutive  even  integers  such  that  3  times 
the  first  is  12  greater  than  the  third. 

14.  Find  two  consecutive  integers  such  that  3  times  the  first 
plus  7  times  the  second  equals  217. 

15.  Find  two  consecutive  integers  such  that  7  times  the 
first  plus  4  times  the  second  equals  664. 

16.  Find  four  consecutive  odd  integers  such  that  7  times 
the  first  equals  5  times  the  last. 

17.  Find  the  side  of  a  square  whose  perimeter  is  64  inches. 

18.  A  rectangle  is  4  inches  longer  than  it  is  wide.  Find  its 
length  and  width  if  the  perimeter  is  40  inches. 

19.  A  rectangle  is  twice  as  long  as  it  is  wide.  Find  its 
dimensions  if  the  perimeter  exceeds  the  length  by  60. 

20.  The  length  of  a  rectangle  is  \\  times  as  great  as  its 
width.  Find  its  dimensions  if  the  perimeter  exceeds  the 
width  by  40  inches. 

21.  The  width  of  a  rectangle  is  \  of  its  length  and  the  pe- 
rimeter exceeds  the  length  by  50  inches.     Find  its  dimensions. 

22.  The  melting  point  of  iron  is  450  degrees  centigrade 
higher  than  5  times  that  of  tin.  Three  times  the  number  of 
degrees  at  which  iron  melts  plus  7  times  the  number  at  which 
tin  melts  equals  6410.     Find  the  meking  point  of  each  metal. 


SOLUTION   OF   PROBLEMS  41 

23.  How  many  dollars  will  amount  to  $  620  in  4  years  at 
6  %  simple  interest  ? 

Solution.     From  arithmetic  we  have 

amount  =  principal  +  interest^ 
or  a=p  -\-  i  =p  -^^  prt. 

Hence,  by  the  conditions  of  the  problem, 

620=p  +  .06  X  4  xp. 
By  Principle  IL  620  =  ^9(1  +  .24). 

ByZ),  p  =  ^=500. 

^      1.24 

24.  Find  what  principal  invested  for  6  years  at  4i  %  simple 
interest  will  amount  to  $  1270. 

25.  Find  what  principal  invested  for  12  years  at  5^  %  sim- 
ple interest  will  amount  to  $  4150. 

26.  How  much  must  be  invested  at  6  %  interest  to  amount 
to  $  2650  at  the  end  of  one  year  ? 

27.  How  much  must  be  invested  at  5  %  simple  interest  to 
amount  to  $  2025  at  the  end  of  7  years  ? 

28.  At  what  rate  of  interest  per  year  must  $800  be  in- 
vested to  amount  to  $  1000  in  5  years  ? 

HISTORICAL  NOTE 

Representation  of  Unknown  Numbers.  The  historical  beginnings  of 
algebra  are  found  in  the  attempted  solution  of  problems.  The  earliest 
work  known  on  algebra  is  by  an  Egyptian  priest,  Ahmes,  who  used  "  heap  " 
to  represent  the  unknown.  One  of  his  problems  reads,  "  heap,  its  seventh, 
its  whole,  it  makes  nineteen, "  which  means :  Solve  the  equation  -  +  x  =  10. 

Diophantus,  a  Greek  (300  a.d.),  used  the  letter  s  to  stand  for  the  un- 
known. The  Arabians  used  the  word  "thing"  and  the  early  European 
algebraists  often  used  the  Latin  equivalent  7'es.  Francois  Vieta  used  capi- 
tal letters  such  as  A.  The  fame  of  Vieta  was  such  that  other  writers  were 
led  to  follow  him.  Though  many  writers  before  Vieta  had,  now  and  then, 
used  letters  to  represent  the  unknowns,  we  may  regard  the  final  establish- 
ment of  this  custom  to  date  from  his  work. 

Ren^  Descartes  (see  page  239)  fixed  the  custom  of  using  the  last  letters 
of  the  alphabet  for  the  unknowns. 


42  EQUATIONS   AND   PROBLEMS 

REVIEW  QUESTIONS 

1.  Define  equation;  identity.  State  in  detail  how  the 
equation  and  the  identity  differ.     Give  an  example  of  each. 

2.  What  value  of  x  satisfies  the  equation  a;  -f-  4  =  9  ? 
What  value  of  x  will  satisfy  the  equation  obtained  by  adding 
7  to  each  member  of  this  equation  ?     by  adding  12  ?     24  ? 

3.  If  4  be  added  to  the  first  member  of  the  equation 
X  -{-  4:  =  9,  and  6  to  the  second  member,  what  value  of  x  will 
satisfy  the  equation  thus  obtained  ? 

4.  If  the  same  number  is  added  to  each  member  of  an 
equation,  is  the  resulting  equation  equivalent  to  the  first 
equation  ?     Illustrate  by  an  example. 

5.  If  different  numbers  are  added  to  the  members  of  an 
equation,  is  the  resulting  equation  equivalent  to  the  first  equa- 
tion ?     Illustrate  by  an  example. 

6.  If  the  same  number  is  subtracted  from  each  member  of  an 
equation,  is  the  resulting  equation  equivalent  to  the  first  equa- 
tion ?     Illustrate  by  an  example. 

7.  If  different  numbers  are  subtracted  from  the  members 
of  an  equation,  is  the  resulting  equation  equivalent  to  the  first 
equation  ?     Illustrate  by  an  example. 

8.  Ask  and  answer  questions,  similar  to  the  two  preceding, 
about  multiplying  and  dividing  both  members  of  an  equation  by 
the  same  or  different  numbers.     Illustrate  each  by  examples. 

9.  Each  step  in  solving  an  equation  consists  in  changing  one 
equation  into  another  equivalent  equation  whose  solution  is  more 
apparent  than  that  of  the  original  equation. 

Hence  what  operations  may  be  performed  in   solving   an 
equation  ? 

10.    State  Principle  VI  in  full. 


Franfois  Vieta  (Viete)  (1540-1603)  was  a  French  lawyer  who 
wrote  on  mathematics  for  his  own  amusement.  He  printed  and 
distributed  his  mathematical  papers  at  his  own  expense.  He  used 
letters  systematically  to  represent  numbers,  and  from  his  time  on 
this  became  the  universal  custom  in  algebra. 

Such  characters  as  +.  — ,  though  used  as  early  as  1489  by 
Johann  Widmann,  did  not  come  into  general  use  before  the  time 
of  Vieta.  His  keenness  of  mind  was  once  shown  by  his  discovery 
of  the  key  to  a  Spanish  cipher  consisting  of  more  than  500  char- 
acters. In  their  astonishment  the  Spaniards  appealed  to  the  Pope, 
accusing  Vieta  of  using  Black  Art. 


CHAPTER    III 
POSITIVE   AND   NEGATIVE   NUMBERS 

35.  A  New  Kind  of  Number.  Thus  far  the  numbers 
used  have  been  precisely  the  same  as  in  arithmetic, 
though  their  representation  by  means  of  letters  and 
some  of  the  methods  used  in  operating  upon  them  are 
peculiar  to  algebra. 

We  now  proceed  to  the  study  of  a  new  kind  of  number. 

Examples.  What  is  the  highest  temperature  you 
have  ever  seen  recorded  on  the  thermometer  ?  the 
lowest  ? 

In  answering  these  questions  you  not  only  give  cer- 
tain numbers,  but  you  attach  to  each  a  certain  quality. 
The  temperature  is  above  zero  or  below  zero ;  that  is, 
the  degrees  on  the  thermometer  are  measured  in  op- 
posite directions  from  a  starting  point  which  is  marked 
zero. 

Many  other  pairs  of  quantities  possess  such  opposite 
qualities ;  for  instance,  motion  to  the  right  and  to  the 
left,  gain  and  loss,  credit  and  debit. 

It  has  been  found  useful  in  mathematics  to  extend 
the  number  system  of  arithmetic  so  as  to  make  it  apply 
directly  to  cases  like  these.  The  opposite  qualities 
involved  are  designated  by  the  words  positive  and 
negative. 

It  is  commonly  agreed  to  call  above  zero  positive  and  below 
zero  negative  ;  motion  to  the  right  positive  and  to  the  left  nega- 
tive ;  credit  positive  and  debit  negative  ;  gain  positive  and  loss 
negative. 

43 


44  POSITIVE   AND   NEGATIVE   NUMBERS 

36.  Positive  and  Negative  Numbers.  The  signs  +  and  ~  stand 
respectively  for  the  words  positive  and  negative,  and  numbers 
marked -with  these  signs  are  called  positive  and  negative  numbers 
respectively.     See  §  46. 

Thus,  5'^  above  zero  is  written  +6°,  and  15°  below  zero  is  written  -15°. 

When  no  sign  of  quality  is  written,  the  positive  sign  is 
understood. 

E.g.  +5°  is  usually  written  5°. 

Positive  and  negative  numbers  are  sometimes  called  signed 
numbers,  because  each  such  number  consists  of  a  numerical 
part,  together  with  a  sign  of  quality  expressed  or  understood. 

The  numerical  part  of  a  signed  number  is  called  its  absolute 
value. 

Thus,  the  absolute  value  of  +3  and  also  of  -3  is  3. 

37.  Graphic  Representation  of  Signed  Numbers.  The  integers 
of  arithmetic  may  be  arranged  in  a  series  beginning  at  zero 
and  extending  indefinitely  toward  the  right. 

Thus,  0,  1,  2,  .3,  4,  5,  6,  7,  8,  9,  ••• 

71i€  integers  of  algebra  may  be  arranged  in  a  series  beginning 
at  zero  and  extending  indefinitely  both  to  the  right  and  the  left. 
Thus,   •••  -5,  -4,  -3,  -2,  -1,  0,  +1,  +2,  +3,  +4,  +5,  ■••. 

One  of  the  most  extensive  uses  of  signed  numbers  is  for 
marking  the  points  on  a  straight  line.  On  an  unlimited 
straight  line  mark  any  point  zero.  On  both  sides  of  this  point 
lay  off  equal  divisions,  as  shown  in  the  figure  on  page  45. 

In  order  to  describe  the  position  of  any  one  of  these  division  points^ 
we  need  not  only  an  integer  of  arithmetic,  to  specify  hov!  far  the  given 
point  is  from  the  point  marked  zero,  but  also  a  sign  of  quality  to  indicate 
on  which  side  of  this  point  it  is. 

E.g.  +G  marks  the  division  point  6  units  to  the  riixht  of  zero,  and  —5 
marks  the  point  5  units  to  the  left  of  zero.  Such  a  diagram  is  called  the 
scale  of  signed  numbers. 

Fractions  would  of  course  be  represented  by  points  between  the  inte- 
gral division  points. 


ADDITION   OF   SIGNED   NUMBERS 


45 


ADDITION  OF   SIGNED   NUMBERS 

38.  Addition  by  Counting.  In  arithmetic  two  numbers  may 
be  added  by  starting  with  one  number  and  counting  forward 
as  many  units  as  there  are  units  in  the  other  number. 

E.g.     To  add  3  to  5  we  start  with  5  and  count  6,  7,  8. 

Two  signed  numbers  are  added  in  the  same  manner  except 
that  the  direction,  forward  or  backward,  in  which  we  count,  is 
determined  by  the  sign  +  or  ~,  of  the  number  which  we  are 
adding. 


.;^8    -7    -fi 

-M — I — y- 


4    -H    -2    -1        0    +1    +2    +3    +4    +5     +6    +7     +8.«. 


Thus,  to  add  +3  to  +5  begin  at  +5  and  count  3  to  the  right. 

To  add  -3  to  +5  begin  at  +6  and  count  3  to  the  left. 

To  add  -3  to  -5  begin  at  -5  and  count  3  to  the  left. 

To  add  +3  to  -5  begin  at  ~5  and  count  3  to  the  right. 

The  results  are  as  follows  : 

+5  +  +3  =  +8  ;         +5  +  -3  =  +2  ;         -5  +  -3  =  -8  ;         -  5  -(-  +3  =  "2. 

+5 +-3  =  +2   is  read  positive  5  plus  negative  3   equals  positive  2. 

-6  +  -3  =  -8  is  read  negative  5  plus  negative  3  equals  negative  8. 

In  like  manner,  read  the  other  two. 


ORAL  EXERCISES 

Using  the  number  scale  above  perform  the  following  additions 
by  counting. 

-2  +  -3. 

-3  +  -5. 
-4  4-  +6. 

-5  -f  +5 
-7  +  +8 
-6  +  +5 
-6  4-  -2 
-7  + +7 
-8 +  +4 
-8-h+6 


1. 

+3  +  -1.  • 

11 

2. 

+4  -f  -2. 

12 

3. 

+5  +  ~3. 

13 

4. 

+5  +  -5. 

14 

5. 

+5  +  -7.   . 

15 

6. 

+5  +  -8. 

16 

7. 

+8  +  -7. 

17 

8. 

+6  -h  -4. 

18 

9. 

+7  +  -7. 

19 

0. 

+9  +  -5. 

20 

21. 

-4 +  -3. 

22. 

-3  +  +7. 

23. 

+8  +  -9. 

24. 

+5  4-  -3. 

25. 

+7  +  -5. 

26. 

+8  +  --7. 

27. 

-3  +  -4. 

28. 

-1  +  ^7. 

29. 

-2  +  -5. 

30. 

-7  +  +5. 

46  POSITIVE   AND   NEGATIVE   NUMBERS 

39.  Further  Illustrations  of  Signed  Numbers.  The  meaning 
of  positive  and  negative  numbers  is  further  explained  in  the 
following 

Illustrative  Problems:  1.  If  a  man  gains  $1500  and  then 
loses  $  800,  what  is  the  net  result?     Answer,  $700  gain. 

In  this  case  the  result  is  obtained  by  subtracting  800  from  1500.  Yet 
this  is  not  really  a  problem  in  subtraction  but  in  addition.  That  is,  we 
are  not  asking  for  the  difference  between  $  1500  gain  and  $  800  loss,  but 
for  the  net  result  when  the  gain  and  the  loss  are  taken  together,  or  the 
sum  of  the  profit  and  the  loss.  Hence,  we  say  $  1500  gain  +  $  800  loss  = 
f  700  gain,  or  using  positive  and  negative  signs, 

+  1500  + -800=  +700. 

2.  The  assets  of  a  commercial  house  are  $250,000,  and  the 
liabilities  are  $275,000.  What  is  the  net  financial  status  of 
the  house  ?     Answer^  $  25,000  net  liabilities. 

Thus, 

$250,000  assets  +  $275,000  liabilities  =$25,000  net  liabilities. 
Or  +250,000  +  -275,000  =  -25,000. 

3.  The  thermometer  rises  18  degrees  and  then  falls  28 
degrees.  What  direct  change  in  temperature  would  produce 
the  same  result  ?     Ansiver,  10  degrees  fall. 

Thus,  18°  rise  +  28^^  fall  =  10°  fall. 

Or  +18 +-28  =  -10. 

4.  A  man  travels  700  miles  east  and  then  400  miles  west. 
What  direct  journey  would  bring  him  to  the  same  final  desti- 
nation ?     Answer,  300  miles  east. 

Thus,  700  miles  east  +  400  miles  west  =  300  miles  east. 

Or  +700  +  -400  =  +300. 

Positive  and  negative  numbers  may  be  written  in  columns 
and  added.     Thus  the  above  examples  would  stand  as  follows  : 

+1500  +250,000  +18  +700 

-800  -275,000  -28  -400 

+700  -26,000  -10  +300 


ADDITION   OF    SIGNED   NUMBERS  47 

The  preceding  exercises  illustrate 

Principle  VII 

40.  Rule.  To  add  two  nujnbers  with  like  signs,  find 
the  sum  of  tlieir  absolute  values,  and  prefix  to  this 
their  cominon  sign. 

To  add  two  numbers  with  opposite  signs,  find  the  differ- 
ence of  their  absolute  values,  and  prefix  to  this  the  sign  of 
that  one  whose  absolute  value  is  the  greater. 

In  case  the  signs  of  two  numbers  are  opposite  and  their 
absolute  values  are  equal,  their  sum  is  zero. 

41.  Signed  Numbers  which  Cancel  each  other.  We  have  seen 
that  in  linding  the  sum  of  two  numbers  whose  signs  are 
opposite,  they  tend  to  cancel  each  other. 

Thus,  in  adding  +5  and  "8,  the  +5  cancels  ~ )  out  of  "8  and  leaves  -.3 
as  the  sum.  In  adding  "5  and  +8,  the  "5  cancels  +5  and  leaves  +3  as  the 
sum. 

ORAL   EXERCISES 

In  this  manner  find  the  sums  of  the  following : 

1.    +7             7.  -8  13.     +7  19.  -25         25.     "9 

4  ^;3 -19  +35  -12 

9  8.   -9  14.   -14  20.   -15 

+24 


+- 


-3 

8. 

-9 

-4 

9. 

-4 

+8 

10. 

+9 

-15 

11. 

+13 

-17 

12. 

+12 

-14 

+12  10.     +9  16.  +20  22.  +20 

-7  -15  -50  -50 


23.  +14 
+12  -17  -25  -21 


14. 

-14 

+8 

15. 

+30 

-40 

16. 

+20 

-50 

17. 

+15 

-25 

18. 

-^10 

-20 

21.  -12 
-18 


12         12.  +12  18.   -"10  24.   +18 

7  -14  -20  -24 


26. 

-12 

-8 

27. 

+12 

-16 

28. 

+18 

-12 

29. 

+20 

-35 

30. 

+35 

-20 

48 


POSITIVE   AND   NEGATIVE   NUMBERS 


Algebraic  Sum.  The  sum  of  signed  numbers  obtained  as  in 
the  preceding  exercises  is  called  their  algebraic  sum. 

Signed  numbers  find  application  in  any  situation  where 
opposite  qualities  of  the  kind  here  considered  are  present. 
Besides  those  already  mentioned,  other  instances  occur  in  the 
applications  below. 

WRITTEN  EXERCISES 

1.  A  balloon  which  exerts  an  upward  pull  of  460  pounds  is 
attached  to  a  car  weighing  275   pounds.     What   is   the    net 

upward  or  downward  pull  ?  Express  this 
as  a  problem  in  addition,  using  positive 
and  negative  numbers. 

Solution.  460  lbs.  upward  pull  plus  275  lbs. 
downward  pull  equals  185  lbs.  net  upward  pull. 
Using  positive  numbers  to  represent  upward  pull 
and  negative  numbers  to  represent  downward  pull, 
this  equation  becomes 

+460  +  -275  =  +185. 

In  each  of  the  following  translate  the 
solution  into  the  language  of  algebra  by 
means  of  signed  numbers  as  in  Example  1. 

2.  A  balloon  which  exerts  an  upward  pull  of  600  pounds 
has  a  450-pound  weight  attached  to  it.  What  is  the  net  up- 
ward or  downward  pull  ? 

3.  A  man's  property  amounts  to  $45,000  and  his  debts  to 
$52,000.     What  is  his  net  debt  or  property  ? 

4.  The  assets  of  a  bankrupt  firm  amount  to  $245,000  and  the 
liabilities  to  $325,000.     What  are  the  net  assets  or  liabilities? 

5.  A  man  can  row  a  boat  at  the  rate  of  6  miles  per  hour. 
How  fast  can  he  proceed  against  a  stream  flowing  at  the  rate 
of  2i  miles  per  hour?    against  one  flowing  7  miles  per  hour? 

6.  A  steamer  which  can  make  12  miles  per  hour  in  still 
water  is  running  against  a  current  flowing  15  miles  per  hour. 
How  fast  and  in  what  direction  does  the  steamer  move  ? 


AVERAGES   OF   SIGNED   NUMBERS  49 

42.  Averaging  Signed  Numbers.  Half  the  sum  of  two  num- 
bers is  called  their  average.  Thus  6  is  the  average  of  4  and  8. 
Similarly,  the  average  of  three  7iumhers  is  one  third  of  their 
sum,  and  in  general  ihe  average  of  n  numbers  is  the  sum  of  the 
numbers  divided  by  n. 

Find  the  average  of  each  of  the  following  sets : 

1.   10,  12,  14,  16,  18.  2.   7,  9,  11,  13,  15,  8. 

The  average  gain  or  loss  per  year  for  a  given  number  of 
years  is  the  algebraic  sum  of  the  yearly  gains  and  losses  di- 
vided by  the  number  of  years. 

Illustrative  Problem.  A  man  lost  $  400  the  first  year,  gained 
$  300  the  second,  and  gained  $  1000  the  third.  What  was  the 
average  loss  or  gain  ? 

Solution.     (-400  +  +300  +  +1000)  --  3  =+900  -  3  =+300. 
That  is,  the  average  gain  is  §  300. 

WRITTEN   PROBLEMS 

1.  Find  the  average  of  $  1800  loss,  $  3100  loss,  $  6800  gain, 
$  10,800  loss,  and  $  31,700  gain. 

Suggestion.  Add  all  the  positive  members  separately  and  all  nega- 
tive members  separately.     Then  combine  the  two  sums. 

2.  Find  the  average  of  S180  gain,  $360  loss,  $480  loss, 
$  100  gain,  $  700  gain,  $  400  gain,  $  1300  loss,  $  300  gain, 
$  4840  gain,  and  $  12,000  gain. 

Find  the  average  yearly  temperatures  at  the  following  places, 
the  monthly  averages  having  been  recorded  as  given  below : 

3.  For  New  York  City  :  +29°,  +33°,  +39°,  +46°,  +53°,  +63°, 
+67°,  +67°,  +61°,  +52°,  +47°,  +41°. 

4.  For  St.  Vincent,  Minnesota:  "5°,  0°,  +15°,  +35°,  +55°, 
+60°,  +66°,  +63°,  +bb°,  +40°,  +22°,  +5°. 

5.  For  Nerchinsk,  Siberia :  "23°,  "13°,  "10°,  +35°,  +55°,  +70°, 
+70°,  +64°,  +50°,  +30°,  +5°,  "15°. 


60  POSITIVE   AND   NEGATIVE   NUMBERS 

SUBTRACTION  OF   SIGNED  NUMBERS 

43.  Subtraction  is  the  process  of  finding  the  number  which 
added  to  the  subtrahend  will  equal  the  minuend. 

Thus,  to  subtract  3  from  8  is  to  find  the  number  which  added  to  3  will 
make  8.     That  is,  8  —  3  =  5  because  3  +  5  =  8. 

Then  the  test  for  subtraction  is 

Remainder  +  Subtrahend  =  Minuend. 

In  subtracting  signed  numbers  we  may  start  on  the  scale  at 
the  point  indicated  by  the  subtrahend  and  iind  hoiv  far  and  in 
which  direction  we  must  count  in  order  to  reach  the  minuend. 

•  r.S    -7-6    -5    -4    -3    -2    -1        0    +1    +'2    +3    +4    +5     +6    +7     +8.«« 

<— 1 \ 1 1 \ 1 \ i \ \ \ 1 \ 1 \ 1 h-^ 

Thus  +8  —  +-5  =  +3,  because  from  +5  to  +8  is  3  in  the  positive  direction. 
+8  —  -5  =  +13,  because  from  ~5  to  +8  is  13  in  the  positive  direction. 
~8  —  ~5  =  ~3,  because  from  ~5  to  ~8  is  3  in  the  negative  direction. 
~8  —  +5  =  ~13,  because  from  +5  to  ~8  is  13  in  the  negative  direction. 

Similarly,  perform  the  following  subtractions : 

1.  -4 --2.  3.    -8 --4.  5.    8 --8.  7.    "7  -  "4. 

2.  -5  -  +6.  4.    7  -  -6.  6.    -4  -  -7.  8.    "3  -  "8. 

44.  A  Short  Rule  for  Subtraction.  Since  +S—'5=  +13,  and 
since  +8  -f-  +5  =  +13,  it  follows  that  subtracting  ~o  from  +8 
gives  the  same  result  as  adding  +5  to  +8.  Similarly,  ~8  —  ~5 
=  -3  and  S  + +5  =  "3. 

Hence,  subtracting  a  negative  number  is  equivalent  to  adding 
a  ])ositive  number  with  the  same  absolute  value. 

Since  +8  —  +5  =  +3,  and  since  +8  -f  "5  =  +3,  it  follows  that 
subtracting  +5  from  +8  gives  the  same  result  as  adding  ~5  to  "*"8 
Similarly,  "8  -  +5  =  "13  and  "8  +  -;">  =  "13. 

Hence,  subtracting  a  j^ositioe  number  is  equivalent  to  adding 
a  negative  number  with  the  same  absolute  value. 

These  statements  are  illustrated  by  such  facts  as :  Removing 
a  debt  is  equivalent  to  adding  property  and  removing  property  is 
equivalent  to  adding  debt. 


SUBTRACTION   OF   SIGNED   NUMBERS  51 

Positive  and  negative  numbers  may  be  written  in  columns 
and  subtracted. 

Thus         +8  +8  -8  -8 

+5  -5  -5  +5 


+3  +13  -3  -13 

ORAL  EXERCISES 

Perform  the  following  subtractions  by  changing  the  sign  of 
the  subtrahend  and  adding : 

1.    -5  3.    +3  5.    -16  7.      16  9.    n6 

-2  -5  -12  +4  -12 


2.    -4  4.    +57  6.    +12  8.    -19  10.    "48 

+1 -32  +50  -24  ^ 

The  preceding  exercises  illustrate 

Principle  VIII 

45.  Rule.  To  subtract  one  sigjied  number  from  another 
signed  number,  change  the  sign  of  the  suhtraliencl  and 
then  add  it  to  the  minuend. 

The  change  in  the  sign  of  the  subtrahend  may  be  made  men- 
tally without  rewriting  the  problem.  Check  by  showing  that 
Remainder  +  ^Subtrahend  =  Minuend. 

WRITTEN  EXERCISES 

Perform  the  following  subtractions,  changing  the  signs  of  the 
subtrahends  mentally : 


1. 

-10  -  -o. 

7. 

+6  -  -14. 

13. 

-78  -  -37. 

2. 

-15  -  +5. 

8. 

+7  -  -9. 

14. 

+57  -  +84. 

3. 

+20  -  -15. 

9. 

-11  -  +6. 

15. 

-48  -  -31. 

4. 

+11  -  +3. 

10. 

-21  -  -6. 

16. 

-39  -  -95. 

5. 

-11  -  +5. 

11. 

+93 -+22. 

17. 

-91  -  -3. 

6. 

-17 --20. 

12. 

+17  -  -13. 

18. 

-38  _  +74. 

52  POSITIVE    AND   NEGATIVE   NUMBERS 

Subtraction  always  Possible  in  Algebra.  In  arithmetic  sub- 
traction is  possible  only  when  the  subtrahend  is  less. than,  or 
equal  to,  the  minuend. 

E.g.  In  arithmetic  we  cannot  subtract  5  from  2  since  there  is  no 
positive  number  which  added  to  5  gives  2. 

However,  in  algebra,  by  means  of  negative  numbers  we  can 
as  easily  perform  the  subtraction,  2  minus  5,  as  5  minus  2. 

Thus,  2  —  5  =  -3,  since  -3  +  5  =  2. 

Similarly,  0— +5  =  "5,  since  "5  +  +5  =  0,  and  -1  — +6  =  "7  since 
-7  +  +6  =  -1. 

46.    Double  Use  of  the  Signs  +  and  — .     In  §  36  we  agreed 

that  when  no  sign  of  quality  is  written,  the  sign  +  is  under- 
stood.    Hence  we  may  write  : 

+8  -f  +5  =  8  -f  5.  (1) 

+8  -  +5  =  8  -  5.  (2) 

By  Principle  VII,  w^e  have 

+8  +  -5  =  +8  -  +5  =  8  -  5.  (3) 

By  Principle  VIII,  we  have 

+8  -  -5  =  +8  -f  +5  =  8  +  5.  (4) 

These  examples  show  how  we  may  dispense  with  the  special 
signs  of  quality  +,  or  ~,  as  follows : 

1.  Positive  numbers  are  icritten  without  any  sign  indicating 
quality  except  lohere  special  emphasis  is  desired,  in  lohich  case  the 
sign  +  is  used. 

2.  A  negative  number  when  standing  alone  is  preceded  by  the 
sign  — .     Thus  ~5  is  written  —  5. 

3.  Wlien  a  negative  number  is  combined  ivith  other  numbers, 
its  quality  is  indicated  by  the  sign  —  with  parentheses  inclosing  it. 

Thus,  8  +  -5  is  written  8  +  (  —  5), 

and  8  —  -5  is  written  8  —  (  —  5). 

But  in  such  cases  it  is  customary  to  apply  Principles  VII  and 
VIII  and  write  at  once  8  —  5  instead  of  8  +  (  —  5)  and  8  +  5 
instead  of  8  —  (  —  5). 


SUBTRACTION   OF   SIGNED   NUMBERS  53 

WRITTEN  EXERCISES 

Work  Examples  1-16  twice,  first  adding,  then  subtracting. 


-25 

5. 

-28 

9. 

13 

13. 

16 

14 

-10 

-20 

30 

25 

6. 

28 

10. 

-13 

14. 

-16 

-14   ' 

-10 

-20 

30 

25 

7. 

-28 

11. 

-13 

15. 

-16 

14 

10 

20 

-30 

-25  . 

8. 

28 

12. 

13 

16. 

16 

-14 

10 

20 

-30 

Perform  the  operations  indicated. 

17.  20- (-5)  22.   15a- (-20a) 

18.  20 +  (-5)  23.    16  71- 25  ?i 

19.  20  -  (  -  30)  24.    40  s  -  (-  30  s) 

20.  20x-30a;  25.    156-356 

21.  15a  — 20a  26.   43a;— 71a; 

EXPLANATORY  NOTE 

On  the  Double  Use  of  Signs.  It  is  clear  from  the  examples  in  this 
chapter  that  signed  numbers  are  needed  to  represent  actual  conditions  in 
life,  as  in  case  of  the  thermometer.  While  from  now  on  such  numbers 
will  be  distinguished,  in  accordance  with  universal  custom,  by  the  signs 
+  ,  — ,  it  should  be  understood  that  each  of  these  signs  is  thus  made  to 
represent  either  one  of  two  entirely  different  things,  namely,  an  operation 
or  a  quality. 

After  we  acquire  some  understanding  of  the  matter,  this  double  use  of 
the  signs  seldom  leads  to  confusion,  since  we  can  always  tell  from  the 
context  which  use  is  meant.  For  example,  in  5  —  3,  the  sign  —  means 
subtraction,  while  in  x  =  —  3  it  means  negative. 

But  for  the  sake  of  avoiding  confusion  at  the  outset,  and  to  make  clear 
that  a  negative  number  is  not  necessarily  a  subtrahend  and  that  a  positive 
number  is  not  necessarily  an  addend,  we  have  up  to  this  time  used  the 
special  signs  +,  -,  which  could  be  readily  distinguished  from  the  signs  of 
addition  and  subtraction.    They  will  now  be  discontinued. 


54  POSITIVE   AND   NEGATIVE   NUMBERS 

WRITTEN   EXERCISES 

Perform  the  following  indicated  operations  : 

1.  Find  the  value  of  a  +  6  if  (1)  a  =  4,  6  =  —  5  ;  (2)  a  =  -  2, 
6  =  _  7  ;  (3)  a  =  -  6,  6  =  8  ;  (4)  a  =  6,  6  =  -  10. 

2.  Find  the  value  of  a  -  6  if  (1)  a  =  8,  6  =  8 ;  (2)  a  =  -  3, 
h  =  -l;  (3)  a  =  4,  6=  -  9  ;  (4)  a  =  -  3,  5  =  0. 

3.  Find  the  value  oi  a  -\- h  -\-  c  \i  (1)  a  =  3,  6  =  —  4,  c  =  —  1 ; 
(2)  a  =  -  7,  ?>  =  3,  c  =  2 ;  (3)  a  =  -  1,  &  =  -  8,  c  =  10. 

4.  Find  the  value  of  a  -f  5  —  c  if  (1)  o  =  6,  6  =  —  4,  c  =  5  ; 
(2)  a  =  -  2,  6  =  -  4,  c  =  -  6  ;  (3)  a  =  7,  6  =  -  8/  c  =  -  6. 

5.  Find  the  value  of  a  —  &  —c  if  (1)  a  =  3,  6  =  6,  c  =  —  2  ; 
(2)  a  =  -  6,  6  =  7,  c  =  -  12  ;   (3)  a  =  8,  ^  =  4,  c  =  8. 

6.  Find  the  value   of    —  a  —  h  -^  c  if    (1)   a  =  7,  h  =—  G, 
c  =  4 ;   (2)  «  =  -  2,  6  -  -  6,  c  =  2. 

7.  Find  the  value  of   —  a  +  ?>  —  c  if  (1)  a  =  —  1,  6  =  —  2, 
c  =  -  3 ;  (2)  a  =  -  8,  6  =  10,  c  =  -  2. 

8.  Find  the  value  of  —  a  —  Z>  —  c  if  (1)  a  =  —  3,  6  =  —  2, 
c  =  -  1 ;  (2)  a  =  6,  6  =  -  3,  c  =  -  9. 

Solve  the  following  equations  : 

9.  ^'  _|_  8  —  4.  Suggestion.     Subtract  8  from  each  member,  or 
-  ^      ,   I    Q -           transpose  8  and  change  its  sign. 

11.  .T  — 9  =  1.  19.  17+.f  =  -35. 

12.  3  4-a;  =  0.  20.  a;- 14  =—18. 

13.  it-  4- 13  =  7.  21.  x-2b  =  16. 

14.  —  4  +  .f  =  —  9.  22.  X  -f  4  =  1. 

15.  —  r>  +  .v  =  4.  23.  X  —  7  =  -  15. 

J  6.     -  5  +  .r  =  1 2.  24.    —  21  4-  ic  =  —  IT. 

17.  -9  +  x  =  -18.  25.    -16+-''=-  18. 

18.  -  35  4-  X  =  17.  26.    -  12  -^  x  =  -  20. 


ADDITION   AND   SUBTRACTION  55 


ORAL  EXERCISES 

Add  the  following  : 
1.         18        4.-9        7.         81        10.         IG        13.    -    9 


-    7  18  -  72  -  24 


2.-6  5.-24  8.         46  11.    -16  14.    -    8 
5_  17              -  38  -    7               14 

3.    _    7  6.-18  9.    -36  12.    -    8  15.    -17 

-14  42                  24  -    9  4 


Subtract  the  following : 

16.        30 

19.          4 

22. 

7 

25. 

-10 

28. 

10 

-15 

-12 

- 

-    7 

20 

-40 

17.    —    4       20.  C       23.     —  20        26.     —  10         29.         70 

8  -    8  14  -20  -40 


18.     —    4       21.     —     7        24.  10        27.     —  lr>         30.     —16 

-12  7  -20  15  8 


Solve  each  of  the  following,  using  an  equation  involving 
positive  and  negative  numbers. 

31.  A  dove  which  can  fly  40  miles  per  hour  in  calm  weather 
is  flying  against  a  hurricane  blowing  at  the  rate  of  60  miles  per 
hour.     How  fast  and  in  what  direction  is  the  dove  moving  ? 

32.  If  of  two  partners,  one  loses  $  1400  and  the  other  gains 
$  3700,  what  is  the  net  result  to  the  firm  ? 

33.  A  man's  income  is  $  2400  and  his  expenses  $  1500  per 
year.     What  is  the  net  result  for  the  year  ? 

34.  A  man  loses  $  800  and  then  loses  $  600  more.  What  is 
the  combined  loss  ?  Indicate  the  result  as  the  sum  of  two  nega- 
tive numbers. 


56  POSITIVE    AND   NEGATIVE   NUMBERS 

MULTIPLICATION   OF   SIGNED   NUMBERS 

47.  The  multiplication  of  signed  numbers  is  illustrated  by 
the  following  problems. 

Illustrative  Problem.  A  balloonist,  just  before  starting,  makes 
the  following  preparations :  (a)  He  adds  9000  cubic  feet  of 
gas  with  a  lifting  power  of  75  pounds  per  thousand  cubic  feet. 
(6)  He  takes  on  8  bags  of  sand,  each  weighing  15  pounds. 
How  do  these  operations  affect  the  buoyancy  of  the  balloon  ? 

Solution,  (rt)  A  lifting  power  of  7o  pounds  is  indicated  by  +75,  and 
adding  such  a  power  9  times  is  indicated  by  +9.  Hence,  +9(+75) 
=  +  675,  or  675  is  the  total  lifting  power  added. 

(5)  A  weight  of  15  pounds  is  indicated  by  —  15,  and  adding  8  such 
weights  is  indicated  by  +  8.  Since  the  total  weight  added  is  120  pounds, 
we  have  +  8(—  15)  =  —  120,  which  is  the  total  depressing  power  added. 

Illustrative  Problem.  During  the  course  of  his  journey  the 
balloonist  opens  the  valve  and  allows  2000  cubic  feet  of  gas  to 
escape,  and  later  throws  overboard  4  bags  of  sand.  How  does 
each  of  these  operations  affect  the  buoyancy  of  the  balloon  ? 

Solution,  (a)  The  gas,  being  a  lifting  power,  is  positive,  but  the 
removal  of  2000  cubic  feet  of  it  is  indicated  by  —  2,  and  the  result  is  a 
depression  of  the  balloon  by  150  pounds  ;  that  is,  —  2  •  (+  75)  =  —  150. 

(6)  The  removal  of  4  weights  is  indicated  by  —  4,  but  the  weights 
themselves  have  the  negative  quality  of  downward  pull.  Hence  to  re- 
move 4  weights  of  15  pounds  each  is  equivalent  to  increasing  the  buoyancy 
of  the  balloon  by  60  pounds  ;  that  is,  —  4  •  (  —  15)  =  +  60  =  60. 

Notion  of  Multiplication  Extended.  These  examples  illustrate 
a  natural  extension  of  multiplication  in  arithmetic. 

E.g.  Just  as3-4  =  4  +  4+4  =12,  so  3  •  (-4)  =  -  4+(-4)  +  (-4) 
=  —  12.     Hence  we  write  +  3  •  (+  4)  =  +  12  =  12,  and  +  3  •  -  4  =  -  12. 

Again,  just  as  we  take  the  multiplicand  additively  when  the 
multiplier  is  a  positive  integer,  co  we  take  it  subtractively  when 
the  multiplier  is  a  negative  integer. 

E.g.     -3.(+4)  =  -(+4)-(+4)-  (+4)  =-12, 
and  -3-  (-4)  =  -(-4)  -(-4)- (-4)=-  (-12)  =  +  12. 

Hence  we  write  —  3  •  ( +  4)  =  -  12  and  — 3  •  ( -  4)  =  +  12  =  12. 


MULTIPLICATION   OF   SIGNED   NUMBERS  57 

Two  numbers   may  be  conveniently  placed  in  columns  for 

multiplying. 

Thus,         +4  +4  — 4  -4 

+  3  -3  +3  -3 

+  12  -12  -  12  +12 

ORAL  EXERCISES 

Explain  the  following  indicated  multiplications  and  find  the 
product  in  each  case. 

1.  _3.(_10).  4.    _7.(-8).  7.    -5.  (-12). 

2.  10.  (-3).  5.   12.  (-4),  8.    -5.  (-8). 

3.  10.  (-5).  6.    -7.  (-6).  9.-8-6 

The  preceding  exercises  illustrate 

Principle  IX 

48.  Rule.  //  two  numbers  have  the  same  si£n,  their 
product  is  positive;  if  they  have  opposite  signs,  their 
product  is  negative. 

In  applying  this  principle  observe  that  the  sign  of  the 
product  is  obtained  quite  independently  ot  the  absolute  value 
of  the  factors. 

E.g.     f-(-5)  =  -0^)  =  -3f;  -  12.(-3i)=  +  42=42. 


ORAL 

EXERCISES 

Ml 

iltiply 

the  following : 

1. 

7 

5. 

6 

9. 

9 

-    4 

-    8 

-10 

2. 

-    5 

6. 

9 

10. 

8 

6 

-    8 

^   6 

3. 

8 

7. 

10 

11. 

-   5 

-    5 

-   8 

4 

4. 

-12 

8. 

20 

12. 

-    6 

4 

6 

5 

58  POSITIVE   AND   NEGATIVE   NUMBERS 

WRITTEN  EXERCISES 

Multiply  the  following  pairs  of  numbers : 


-14 
-   3 

6. 

7. 

8. 

9. 

10. 

45 
6 

25 
4 

-  25 

4 

-  25 

-  4 

-4.2 
6 

11. 
12. 
13. 
14. 
15. 

-5.4 
-    4 

-14 
3 

15  a 
-    4 

14 
-   3 

-22  a; 
-    3 

-40 
5 

-18  m 
4 

-35 
-    5 

-16ab 

-   2 

49.  The  product  of  several  signed  numbers  is  found  as  illus- 
trated in  the  following : 

-2.5.  (-3)  •  (-4)  .  6=  -10  .  (-3)  .  (-4) .  6  =  30  •  (-4)  •  6 
=  — 120  •  6  =  —  720.  That  is,  the  first  two  factors  are  multi- 
plied together,  then  their  product  is  multiplied  by  the  next 
factor,  and  so  on,  until  all  the  factors  are  multiplied. 

Since  the  product  of  all  positive  factors  is  positive,  the  final 
sign  depends  upon  the  number  of  negative  factors.  If  this 
number  is  even,  the  product  is  positive  ;  if  it  is  odd,  the  product 
is  negative. 

E.g.  If  there  are  3  negative  factors,  the  product  is  negative  ;  if  there 
are  4,  it  is  positive. 

In  the  following  exercises  determine  the  sign  of  the  product 
before  finding  its  absolute  value. 

ORAL  EXERCISES 

1.  _4.3.(-2).(-2).  4.    -5. (-4). 3. (-2). 

2.  _2.(-3).(-5).3.  5.    8.(-9).(-l).(-2). 

3.  _5.(-3)(-2)(-4)(-l).     6.    -.5.(-2).(-3).(-4). 


DIVISION   OF   SIGNED   NUMBERS  59 

DIVISION  OF  SIGNED   NUMBERS 

50.  Test  for  Division.  In  arithmetic  we  test  the  correctness 
of  division  by  showing  that  the  quotient  multiplied  by  the  di- 
visor equals  the  dividend. 

E.g.     27  ^  9  :=  3,  because  9  •  3  =  27. 

Hence  division  may  be  defined  as  the  process  of  finding  one  of 
two  factors  when  their  product  and  the  other  factor  are  given. 

The  given  product  is  the  dividend,  the  given  factor  is  the  divi- 
sor, and  the  factor  to  be  found  is  the  quotient. 

In  dividing  signed  numbers  the  above  test  determines  the 
sign  of  the  quotient  as  well  as  its  absolute  value. 

E.g.  -  42  ^ ( 4-  6)  =  -  7,  because  -  7  •  (  +  6)  =  -  42  ; 

also  —  42  -=-(—  6)  =  +  7,  because  +  7  •  (-  6)  =  -  42. 

So  in  every  case  the  test  of  the  correctness  of  division  is : 
Quotient  x  Divisor  =  Dividend. 

Find  the  following  quotients  and  check  as  above: 

1.  —  25-r-5.  3.    5xy^(—x).  5.    75?/h-(— 15). 

2.  -ab^a.  4.    -9nsH-(-3).        6.    -121  a; --11. 

The  preceding  exercises  illustrate 

Principle  X 

51.  Rule.  TJie  qaotient  of  two  signed  niunbersis  posi- 
tive if  the  dividend  and  divisor  have  like  signs,  negative 
if  they  have  opposite  si^ns. 


"  -n 


ORAL 

EXERCISES 

6 
-3 

4. 

-6 
-3 

7. 

-    8 
4 

6 

8 

8. 

-  15 

3 

5. 

-4 

—    5 

-6 

a 

-  8 

p 

-15 

10. 

20 

-10 

11. 

-20 

10 

19, 

-20 

14. 


-48 
-16 

-48 

15.    ^ 

-4  5  -10  16 


60  POSITIVE   AND   NEGATIVE   NUMBERS 

EXERCISBS 
Perform  the  following  indicated  divisions,  and  check  the  first 
ten  by  multiplying  quotient  by  divisor.     Do  Examples  1-21 
orally. 

1.  :^.  8.    ^^ 


7 
42 


-6 

51 

-  17 


9. 

10. 


4.    I^. 


-3 

-75 

5 

-16 
-1 


12. 


13. 


4 

i. 

.(-9). 
-3 

3-8 
-4 

IC 

|.(-9) 

—  5 

3 

.(_4).(- 

-2). 

.6 

-3 

■42  a 
-3 

•42  a; 

15. 

-3 

16. 

75abc 

• 

—  a 

17. 

-lOOxy 

—  x 

18. 

-3202/ 

80 

19. 

25  xy 

—  X 

20. 

— 196  mw 

-14 

oi 

-39  ah 

7.    .  14.  

7  3  -13  a 

22.  A  man  lost  $300,  $500,  and  $700  during  three  consecu- 
tive months.     Express  his  average  monthly  loss  as  a  quotient. 

23.  During  five  consecutive  days  the  minimum  temperature 
was  -  5°,  -  8°,  -  10°,  -  4°,  -  6°  respectively.  Find  the  aver- 
age of  these  temperatures. 

24.  A  trader  lost  $  250  in  each  of  three  months  and  gained 
$75  during  each  of  the  four  succeeding  months.  Find  the 
average  gain  or  loss  for  the  seven  months. 

25.  On  a  cold  day  the  following  temperatures  was  observed : 

—  20°,  -  16°,  -  12°,  -  4°,  2°,  8°,  4°,  -  2°.     Find  the  average 
of  these  readings. 

26.  Find  the  average  of  the  numbers :  280,  —  960,  —  840, 

-  360,  860,  -  260,  - 180,  530,  and  -  480.     See  suggestions 
under  Example  1,  page  49. 


DIVISION   OF   SIGNED  NUMBERS  61 

52.  The  Principles  applied  to  Signed  Numbers.  While  Prin- 
ciples I-V  were  studied  in  connection  with  unsigned,  or  arith- 
metic numbers  only,  it  is  now  important  to  note  that  they  all 
apply  to  signed  numbers  as  well. 

In  the  statement  of  these  principles  the  word  number  will 
from  now  on  be  understood  to  refer  either  to  the  ordinary 
numbers  of  arithmetic  or  to  the  signed  numbers  of  algebra, 
as  occasion  may  require.  It  should  also  be  noticed  that  the 
numbers  of  arithmetic  are  used  as  freely  in  algebra  as  in 
arithmetic.  It  is  only  when  we  wish  to  distinguish  them 
from  negative  numbers  that  they  are  called  positive  numbers. 

ORAL  EXERCISES 

If  a  =  6,  b  =  4,  c  =  —  2,  evaluate  the  following : 

1.  a{b+c).  3.    a(c  —  b).  5.    c(b  —  a). 

2.  a{b  —  c).  4.    c{a  -f  b).  6.    c{a  —  b). 

WRITTEN    EXERCISES 

1.   Find  the  quotient  2(-3)(- 4X- 6)(- 8)(- 2). 
^  4(-12)(-16) 

Solution.  There  are  five  negative  factors  in  the  dividend  and  two  in 
the  divisor.  Hence  the  sign  of  the  dividend  is  —  and  that  of  the  divisor 
is  -f  .  (See  §  49.)  Therefore  the  sign  of  the  quotient  is  —  .  See  §  51. 
We  now  cancel  as  if  all  the  factors  were  positive  and  prefix  the  negative 
sign  to  the  quotient,  obtaining  —  3. 

In  like  manner  find  the  following  quotients : 

2  a(-b)(-e)(-d)  5        ab(- c)(d)(- e) 

ab{-e){-d)  '    {- a){- b)(- d){e) 

3  3(-12)(-18X-32)  x(- y)( -  z)(- A) 

6.8(-24)(-3)  ■  xy{--iz) 

^       _8.9(-3)(-4)  2x(-i/)(-z)(-v) 

_3(-4)(-2)(-6)  •     -xv{-z}y(-3) 


62  POSITIVE   AND   NEGATIVE   NUMBERS 

The  number  system  of  algebra,  so  far  as  we  have  now  studied 
it,  consists  of  the  numbers  of  arithmetic  together  with  the 
negative  numbers. 

HISTORICAL  NOTE 

The  Development  of  Negative  Numbers.  The  Greeks  had  no  concep- 
tion of  a  negative  number  as  distinct  from  a  number  to  be  subtracted. 
Diophantus  states  tliat  in  the  multiplication  of  {a  —  h)  by  (c  —  d),  a 
subtraction  multiplied  by  a  subtraction  gives  an  addition.  That  is. 
{—  b){—  d)=bd.  But  in  applying  tlie  rule  he  takes  care  that  a  is 
greater  than  6,  and  c  greater  than  d.  However,  the  Hindus  appear  to 
have  had  quite  clear  notions  of  a  purely  "negative  number"  as  distinct 
from  a  number  to  be  subtracted.  They  recognized  the  difference  be- 
tween positive  and  negative  numbers  by  attaching  to  one  the  idea  of  debt 
and  to  the  other  that  of  assets,  or  by  letting  them  represent  distances  in 
opposite  directions.  The  Arabs,  however,  failed  to  understand  the  nega- 
tive numbers  and  did  not  include  them  in  the  algebra  which  they  brought 
to  Europe.  (See  page  32.)  Until  the  beginning  of  the  seventeenth 
century,  mathematicians  dealt  almost  exclusively  with  positive  numbers. 

Thomas  Harriot,  an  Englishman  (1560-1621),  was  the  first  to  write  a 
negative  number  all  by  itself.  Thus  an  equation  like  x  =  —  3  was  never 
written  by  any  of  Harriot's  predecessors. 

The  negative  numbers  were  brouglit  permanently  into  mathematics  by 
Ren6  Descartes.  (See  page  239.)  Trying  to  number  all  the  points  of  a 
complete  straight  line,  Descartes  was  compelled  to  start  at  some  point 
and  number  in  both  directions.  Then  it  became  convenient  to  dis- 
tinguish the  numbers  on  the  two  sides  of  this  starting  point  as  positive 
and  negative,  respectively. 

Sir  Isaac  Newton  (see  page  101)  was  the  first  to  let  a  letter  stand  for 
any  number,  negative  as  well  as  positive.  In  such  a  formula  as  a(6  -}-  c)  = 
ab  -\-  ac,  the  predecessors  of  Newton  had  restricted  the  letters  to  represent 
any  positive  numbers,  while  Newton  regarded  the  letters  as  representing 
any  numbers  whatever,  either  positive  or  negative.  This  was  of  very  great 
importance,  since  it  greatly  reduced  the  number  of  formulas  required. 

Negative  numbers  appearcMl  "absurd"  or  "fictitious"  until  a  visual 
or  graphical  representation  of  them  was  discovered.  Cajori  in  his  his- 
tory of  elementary  mathematics  says:  "Omit  all  illustrations  by  lines, 
thermometers,  etc.,  and  negative  numbers  will  be  as  obscure  to  modern 
students  as  they  were  to  the  early  algebraists."  From  the  experience  of 
the  early  mathematicians  it  would  appear  that  if  the  pupil  wishes  really 
to  understand  positive  and  negative  numbers,  he  must  study  with  care 
applications  such  as  are  given  in  the  first  part  of  this  chapter. 


INTERPRETATION   OF   NEGATIVE   NUMBERS  63 

INTERPRETATION   AND    USE   OF   NEGATIVE   NUMBERS 

53.  A  negative  result  obtained  in  solving  a  problem  may 
have  a  natural  interpretation,  or  it  may  indicate  that  the  con- 
ditions of  the  problem  are  impossible. 

A  similar  statement  holds  regarding  fractional  or  zero  answers  in  arith- 
metic, For  example,  if  we  say  there  are  twice  a.s  many  girls  as  boys  in  a 
schoolroom  and  35  pupils  in  all,  the  nuinber  of  boys  would  be  35-h3=11|, 
which  indicates  that  the  conditions  of  the  problem  are  impossible. 

Again,  if  three  buildings  cost  in  all  8  18,500,  antl  if  the  second  cost 
f  9000  more  than  the  first,  and  the  third  $  9500  more  than  the  second, 
then  the  cost  of  the  first  building  would  be  zero,  which  is  impossible. 

Illustrative  Problem.  The  crews  on  three  steamers  together 
number  94  men.  The  second  has  40  more  than  the  first,  and  the 
third  20  more  than  the  second.     How  many  men  in  each  crew  ? 

Solntion.  Let  n  —  number  of  men  in  first  crew. 

Then,  «  +  40  =  number  of  men  in  second  crew, 

and  71  +  40  -f  20  =  number  of  men  in  third  crew. 

Hence,  n  +  n  +  40  -|-  w  +  40  +  20  =  94, 

and  3  ?i  +  100  =  94. 

3n=-6. 
71  =—  2. 
Here  the  negative  result  indicates  that  the  conditions  of  the  problem 
are  impossible. 

Illustrative  Problem.  A  real  estate  agent  gained  S  8400  on 
four  transactions.  On  the  first  he  gained  $  6400,  on  the  sec- 
ond l^e  lost  $  2100,  on  the  third  he  gained  $  5000.  Did  he 
lose  or  gain  on  the  fourth  transaction,  and  how  much? 

Solution.     Since  we  do  not  know  whether  he  gained  or  lost  on  the 

fourth  transaction,  we  represent  the  unknown  number  by  n,  which  may  be 

positive  or  negative,  as  will  be  determined  by  the  solution  of  the  problem. 

Then  we  have  6400  +  ( -  2100)  -f-  5000  +  n  =  8400.  (1) 

Hence,  by  VII,  F,  9300  +  n  =  8400.  (2) 

By  S,  w  =  8400 -9300.     (3) 

By  VIII,  n  =  -  900.  (4) 

In  this  case  the  negative  result  indicates  that  the  conditions  of  the 

problem  are  possible,  and  that  there  was  a  loss  on  the  fourth  transaction. 


64  POSITIVE   AND  NEGATIVE   NUMBERS 

PROBLEMS 

In  the  following  problems  give  the  solutions  in  full  and  state 
all  principles  used. 

In  case  a  negative  answer  is  found,  state  whether  this 
answer  has  a  natural  interpretation,  or  whether  it  indicates 
that  the  conditions  of  the  problem   are  impossible. 

1.  A  man  gains  $2100  during  one  year.  During  the  first 
three  months  he  loses  $  125  per  month,  then  gains  $  500  per 
month  during  the  next  five  months.  What  is  the  average  gain 
or  loss  per  month  during  the  remaining  four  months  ? 

2.  A  man  rowing  against  a- swift  current  goes  9  miles  in  5 
hours.  The  second  hour  he  goes  two  miles  less  than  the  first, 
the  third  three  miles  more  than  the  second,  the  fourth  one  mile 
more  than  the  third,  and  the  fifth  one  mile  more  than  the 
fourth.  How  many  miles  did  he  go  during  each  of  the  five 
hours  ? 

3.  There  are  three  trees  the  sum  of  whose  heights  is  108 
feet.  The  second  is  40  feet  taller  than  the  first,  and  the  third 
is  30  feet  taller  than  the  second.     How  tall  is  each  tree  ? 

In  the  next  three  examples  find  the  average  yearly  tempera- 
tures, the  average  monthly  temperatures  being  as  here  given : 

4.  Port  Conger,  off  the  northwest  coast  of  Greenland  :  —  37°, 
-  43°,  -  32°,  -  15°,  14°,  18°,  35°,  34°,  25°,  4°,  -  17°,  -  30°. 

5.  Franz  Joseph's  Land :  -  20°,  -  20°,  - 10°,  0°,  15°,  30°, 
35°,  30°,  20°,  10°,  0°,  -  10°. 

6.  North  Central  Siberia :  -  60°,  -  50°,  -  30°,  0°,  15°,  40°, 
40°,  35°,  30°,  0°,  -  30°,  -  50°. 

7.  A  merchant  gained  an  average  of  S  2800  per  year  for  5 
years.  The  first  year  he  gained  $  3000,  the  second  $  1500, 
the  third  $  4000,  and  the  fourth  $  2400.  Did  lie  gain  or  lose 
during  the  fifth  year,  and  how  much  ? 


REVIEW   QUESTIONS  65 

8.  A  certain  business  shows  an  average  gain  of  S  4000  per 
year  for  6  years.  During  the  first  five  years  the  results  were : 
S8000  loss,  $10,000  gain,  S  7000  gain,  $3000  gain,  and 
$  12,000  gain.     Find  the  loss  or  gain  during  the  sixth  year. 

9.  A  commercial  house  averaged  $  10,000  gain  for  6  years. 
What  was  the  loss  or  gain  the  first  year  if  the  remaining  years 
show :  $  8000  gain,  $  24,000  gain,  $  2000  loss,  $  20,000  gain, 
and  $  30,000  gain,  respectively  ? 

REVIEW  QUESTIONS 

1.  Name  several  pairs  of  opposite  qualities  all  of  which 
are  conveniently  described  by  the  words  positive  and  nef/atice. 
What  symbols  are  used  to  replace  these  words  Avhen  applied 
to  numbers  ? 

2.  When  loss  is  added  to  profit,  is  the  profit  increased  or 
decreased  ?  What  algebraic  symbols  may  be  used  to  distin- 
guish the  numbers  representing  profit  and  loss  ? 

3.  On  the  number  scale  indicate  what  is  meant  by  -f  2 ; 
by  —  2.  Indicate  what  is  meant  by  the  sign  -|-  in  5  -f  2 ;  by 
the  sign  —  in  5  —  2 ;  by  the  sign  —  in  .t  =  —  2. 

4.  Why  do  we  call  positive  and  negative  numbers  signed 
numbers  ?     What  is  meant  by  the  absolute  value  of  a  number  ? 

5.  State  Principle  YII  in  full. 

6.  How  is  the  correctness  of  subtraction  tested  in  aritli- 
metic  ?     Is  the  same  test  applicable  to  subtraction  in  algebra  ? 

7.  Illustrate  the  subtraction  of  positive  and  negative  num- 
bers by  an  example  involving  profit  and  loss. 

8.  Show  by  counting  on  the  number  scale  that  the  result 
of  subtraction  gives  the  distance  from  subtrahend  to  minuend 
and  that  the  sign  of  the  remainder  shows  the  direction  from 
subtrahend  toward  the  minuend.  For  example,  use  8  — (—5) 
and  —  8  —  (-f  5)  to  illustrate  this. 


66  POSITIVE   AND   NEGATIVE    NUMBERS 

9.    How  do  negative  numbers  make  subtraction  possible  in 
cases  where  it  is  impossible  in  arithmetic  ? 

10.  What  is  a  convenient  rule  for  subtracting  signed  num- 
bers ?     State  Principle  VIII. 

11.  Write  an  equation  whose  solution  is  a  negative  number. 

12.  Give  an  example  in  which  positive  and  negative  num- 
bers are  multiplied.     State  Principle  IX. 

13.  Define  division.  How  do  we  obtain  the  law  of  signs  in 
division  ?  State  Principle  X.  What  is  the  test  of  the  cor- 
rectness of  division  ? 

14.  Explain  how  one  set  of  signs  +  and  —  can  be  used  to 
indicate  both  quality  and  operation. 

15.  By  means  of  Principles  VII,  VIII,  IX,  and  X,  simplify 

ft  ... 

the   expressions,  «-(-&,  a  —  b,  a  -  b,  -,  after   substituting   m 

each  various  positive  and  negative  values  of  a  and  b. 

16.  Add  Principles  VII,  VIII,  IX,  and  X  to  the  list  which 
you  made  in  Chapters  I  and  II.  It  is  absolutely  necessary 
that  you  remember  the  rules  stated  in  these  principles.  Any 
short  phrases  that  will  assist  you  in  this  are  of  value.  For 
instance,  the  following : 

VII.  In  addition,  positive  and  negative  numbers  tend  to  cancel 
each  other.  TJie  common  sign  or  the  sign  of  the  numerically 
greater  is  the  sign  of  the  result. 

VIII.  In  subtraction,  change  the  sign  of  the  subtrahend  and  add. 

IX.  Li  midtiplication,  two  like  signs  give  -h  and  two  unlike 
■  signs  give  — . 

X.  In  division,  like  signs  give  -}-  and  unlike  signs  give  — . 


CHAPTER   IV 

ADDITION   AND    SUBTRACTION   OF    ALGEBRAIC 
EXPRESSIONS 

54.  Building  Algebraic  Expressions.  In  the  preceding  chapter 
we  have  noticed  that  in  solving  problems  we  are  led  to  repre- 
sent numbers  by  means  of  algebraic  expressions  which  are 
formed  by  combining  several  algebraic  symbols. 

E.g.  If  X  is  a  number  representing  my  age  in  years,  then  x—  \0  is 
the  number  representing  my  age  10  years  ago,  and  2(0:—  10)  is  double 
the  number  representing  my  age  ten  years  ago. 

Such  expressions  are  now  to  be  studied  more  in  detail. 

55.  Polynomials ;  Terms.  An  algebraic  expression  composed 
of  parts  connected  by  the  signs  +  and  _  is  called  a  polynomial. 
Each  of  the  parts  thus  connected,  together  with  the  sign  pre- 
ceding it,  is  called  a  term. 

E.g.  5  a  —  3  xj/  —  I  rf  +  99  is  a  polynomial  whose  terms  are  5  «,  —  3  xy, 

—  I  rt,  and  +  99.     The  sign  +  is  understood  before  5  a. 

A  polynomialbf  two  terms  is  called  a  binomial;  one  of  three 
terms  is  called  a  trinomial.  A  term  taken  by  itself  is  called  a 
monomial.     Terms  ^vhicli  are  to  be  added  are  called  addends. 

E.g.  5  a  —  3  xij  is  a  binomial ;  5  a  —  3  xy  —  |  rt  is  a  trinomial  whose 
terms  are  the  monomials  5  a,  —  3  xy.,  —  f  rt. 

According  to  the  above  definition  .r-f  (&  +  c)  may  be  called 
a  binomial  though  it  is  equivalent  to  the  trinomial  x-\-h-\- c. 

In  this  case  x  is  called  a  simple  term  and  (b  +  c)  a  compound 
term.  Likewise  we  may  call  3t  -\-4:X  —  5(a  +  b)j/  a  trinomial 
having   the    simple   terms   3^,  4  aj,  and   the    compound    term 

-  5{a  +  b)y. 

67 


68  ALGEBRAIC   EXPRESSIONS 

56.  Similar  Terms.  Two  terms  which  have  a  factor  in  com- 
mon are  said  to  be  similar  with  respect  to  that  factor. 

E.g.  5  a  and  —3a  are  similar  with  respect  to  a;  —  3  :»•?/ and  —7x 
are  similar  with  respect  to  x  ;  5  a  and  —  5  &  are  similar  with  respect  to  5  ; 
7  ahc  and  —  |  abc  are  similar  with  respect  to  abc. 

Similar  terms  may  be  combined  by  Principle  I. 

JS.g.     o  a  +  3a  =  (5  +  3)a  =  8a  ;    Sxy  —  7  x  =  x{Sy  —  7)  ;    ba  —  5h 

=  5(a  —  h)  ;  ax  +  hx  —  ex  =  {a  -^  h  —  c)x. 

ORAL  EXERCISES 

Select  the  common  factor  and  combine  the  similar  terms  in 
each  of  the  following  : 

1.  5  a  +  3  a.  9.  12  v  -\-  A  v  -\-  6  v. 

2.  4  ?i  H-  5  w.  10.  7  ?>i  +  4  m  +  8  m. 

3.  2  6  +  10  6.  11.  8  a- 3  a +  2  a. 

4.  9c  +  8c  +  7c.           *  12.  3x  +  7x  —  6x. 

5.  14  «  +  7  a;  -f  4  a.\  13.  5  y  —  2  y  —  y. 

6.  2  A:  +  4  ^'  -h  8  A%  14.  5  a6  +  3  a6  -h  2  ab. 

7.  Sy  +  Sy  -h5y.  15.  7ct'  — 5x  +  4a;. 

8.  162  +  2  2 -f  32;.  16.  3a  — 2  a  + 4a. 

WRITTEN  EXERCISES 

Combine  similar  terms  in  the  following : 

17.  ax         20.         3  ab  23.  7  ax         26.        5  ax 

bx  -2ab 

ex  5  ab 

18.  2  ar         21.        11  rs         24. 


2ar 

3  6r 

-2cr 

V2cd 

-4  c/' 

-5gc 

19.        12ca        22.        4a;?/2  25. 


11  rs 

-2s^ 

4  as 

4  a;?/2 
2  xyz 
—  Sxyz 

Sbx 

3  ax 

12  ex 

27. 

2  ax 

6  ab 

4.xy 

7  ac 

-3yz 

--  ad 

—  5wy 

9abe 

28. 

7  xyc 

3dbc 

—  f)  ayx 

—  4e6c 

2bxy 

ADDITION   OF   POLYNOMIALS  ^ 

ADDITION   AND   SUBTRACTION  OF   POLYNOMIALS 
Addition  of  Polynomials.     In  adding  polynomials  we  use 

Principle  XI 

57.  Rule.  If  several  terms  are  to  he  added,  they  may 
he  arranged  and  comhined  in  any  desired  order. 

The  truth  of  this  principle  may  be  seen  from  simple  examples : 

Thus,     2  +  3  +  5  =  3  +  2  +  5  =  2  +  (3  +  •'))  =  (3  +  2)  +  5  =  10. 
Also,    8  +  (-2)  +0  =-2*+8  +  6  =-2  +  (8  +  0)  =  12. 

HISTORICAL  NOTE 

Associative  and  Commutative  Laws  of  Addition.  The  fundamental 
character  of  Principle  XI  was  first  recognized  about  one  hundred  years 
ago.  The  principle  as  here  given  combines  in  one  statement  two  laws 
of  algebra :  (1)  the  associative  law.  first  so  called  by  F.  S.  Servois 
(1814)  ;  (2)  the  commutative  law,  first  so  called  by  Sir  William  Hamilton, 
The  associative  law  states  that  addends  may  be  grouped  in  any  manner. 
Thus,  a  +  &  +  c  =  a  +  (&  +  c)  =  (a  +  5)  +  c. 

The  commutative  law  states  that  addends  may  be  put  in  any  desired 
order.    Thus,  «  +  6  =  5  +  «. 

58.  Arranging  Terms  in  Columns.  In  adding  polynomials  the 
work  may  be  arranged  conveniently  by  placing  similar  terms  in 
the  same  column.     This  is  permissible  by  Principle  XI. 

Example.  Add  5  cc  —  6  ?/  +  4  2  +  5  a^ ;  —  ox  -\-  11  y  —  IQz 
—  ^ht)  and  —  7  ?/  +  8  2;. 

Arranging  similar  terms  in  columns,  and  applying  Principles  I  and 
Vn,  we  have  ^  a      ,     a      ,  c    -. 

-Zx+ny-l(Sz-9ht 

-    7y+    %z  


2x-    2y-    42;  +  (5a-l)6)< 

Check.     Putting  x  =  y=z  =  t  =  a=b  =  l,in  each  of  the  polynomials, 
and  in  their  sum  we  have  : 

5_    6+    4  +  5=        8 
_3  +  ll  _  16_9=-17 

-7+8=        1 

2-2-    4  +  5-9=   -8 


70  ALGEBRAIC   EXPRESSIONS 

WRITTEN   EXERCISES 

Arrange  similar  terms  in  columns  and  add : 

1.  Add  7  5 -3  c  + 2d;   -  2  6  +  8  c  -  13  d. 

2.  Add  6x  —  3y-\-4:t  —  7z;    x—  5y  —  3t',    4cc— 4?/-f8^. 

3.  Add    7  a  —  4:X-{-12z;    8  a  — 3  x +  2  2;     2  a-\-4:X —  Sz; 
5  a  — 2  X  —  4:  z. 

4.  Add    5  ac  +  3  5c  -  14  c  +  8  6;       25 +  3  c- 12  5c  -  3  ac; 
4  5  +  4  c  H-  5c  —  c(c;  2  5c  +  4  ac  4-  c ;  3*5  —  4  c. 

5.  Add    16  xy  —  13  cd;    15ab  —  2  xy;    34:  cd  —  3  xy  -\- 2  ah ; 
14  cd  —  3  xy  —  2  ah. 

6.  Add    34  ax -\- 4:  hy  —  3  z]     2  5?/-}- 5  2;;     3  «a;— 7  5?/ +  5^;; 
1  ax -\- 4,  hy  —  4,  z. 

7.  Add    3a5  +  4cd  — 2ae;  a5  — 3cdH-3ae;    3cd  — 2a5; 
4  cd  —  5  ae  +  7  ah. 

8.  Add  7  ax  —  13  5//  +  5 ;     9  aa;  +  8  5?/  —  4 ;     3  by  —  12  ax ; 
4  ax  +  7  hy  —  9. 

9.  Add   5  a5  -  3  •  07  +  5(x  -  1)  ;    5  •  67  +  3  a5  -  2{x  -  1) ; 
3(a;_l)_4  .67  +2a5. 

10.  Add   ll{G  —  ^)+3{x  +  y)  +  21iDU\    —lliuu  —  lj{x  +  y); 
18  icu  +  2{x  +  2/)  -  13(c  -  9). 

11.  Add5(a  +  5)-3(c-(0;  3(c-d)-8(a  +  5);   -2(a  +  5); 
13(c-d)-  4(a  +  5). 

12.  Add  3H-4(c  — d)— 5(a  — 5  — c);  4(a  —  5  —  c)-|-5(c  —  d)  ; 
3(a  -  5  -  c)-  9(c  -  d)+  12. 

13.  Add   (a-5)-3(c-d)-f  4(a  +  5);    5(a  -  5)  +  4(c  -  d)  ; 
7(c  -  d)-  9(a  -  5)+  3(a  +  5). 

14.  Add    l{x  —  y)  —  4(x  +  y)  +  4  a5  ;        9(.r  +  ?/)  +  3(;r  —  ?/)  ; 
6(.T  -  y)-  2  ah  -  3{x  +  y). 

15.  Add  3(x-5)  +  4(5  +  c)+3(.i'-//);    8(5  +  c-)- r)(x- ?/); 
8(a;  _  5)-  7(5  -h  c)-  4{x  -  7/)  ;  3(x  -  7/)H-(x  -  5). 


I     .'    I  '.    '    ■      'I 


Sir  William  Rowan  Hamilton  (1805-1865)  was  born  in  the 
city  of  Dublin,  of  Scotch  parents.  Already  in  early  childhood  he 
gave  evidence  of  a  brilliant  mind.  As  a  young  man  at  college 
"  amongst  a  number  of  competitors  of  more  than  ordinary  merit 
he  was  first  in  every  subject  and  at  every  examination." 

Before  taking  his  final  examinations  he  was  appointed  in  1827 
to  the  professorship  of  astronomy  in  the  University  of  Dublin. 
He  made  a  profound  study  of  algebra  and  created  a  new  branch 
of  that  subject  which  is  called  quaternions.  In  the  opinion  of  some 
writers  this  will  ultimately  be  regarded  as  one  of  the  great  dis- 
coveries of  the  nineteenth  century. 


ADDITION   OF   POLYNOMIALS  71 

16.  Addl6{a  -^  b  -  c)-  3(x  -  y)+  2{a  -  b);  2{x- y)  -  Axjr, 
S(a  —  b)-^{a-\-b  —  c)  ;  7  (a  —  b)  +  4(x  —  y)  —  8(a  -{-b  —  c). 

17.  AdiHj{a-b)-5{x-^y)-\-7{x-z)-4.abc;  7(x  —  z)-\-5- 
9(x-^y)+(a-b)  +  2  abc ;  ll{a-b)  +  10  abc  +  3(a;-2;)-|-  8(ic+?/). 

18.  Add  2(x  —  y  +  z)+  7(a  —  .-v)  —  3(z  —  y);         3(x—y-\-z) 

-  4(a  -  x)  +  5(2;  -  y) ;  5{a  -  x)  -  2(z  -  y)  +  5(x  -y-i-z); 
5(x  -y-^z)-\-  Mz  -  y). 

19.  AM2ab-{-^{a-b)-\-a(b  +  ?>)  +  b{a+2);  4.ab-2(a-b) 
-h  2  a{b  +  3)  -  2  b{a  +  2) ;  2  6(a  +  2)  +  a{Jj  +  3)  -  2  ab. 

20.  Add  x[y  —  z)  +  y{x  —  z)  -{-  z(x  —  y)  ;  5  xy  —  2  x(y  —  z) 
-\-3z(x-y);   —3xy-\-2x(y~z);    —xy-x(y  —  z)-{-2y(x-z). 

59.  Adding  Polynomials  without  Rewriting  them.  In  practice 
polynomials  may  be  added  without  writing  the  similar  terms 
in-  columns. 

For  example,  to  add   —  Z  a  +  2  h  —  4  c,  5a-!-4  6  +  2c,    and  7  a  —  3  6 

—  5  c,  we  first  pick  out  the  terms  having  a  as  a  factor  and  add  them 
mentally^  then  the  terms  containing  6,  and  finally  those  containing  c  : 

Thus,  — 3a  +  5a+7a  =  9a;  26+46-36  =  36;  -4c  +  2c-5c 
=  -7c. 

Hence  the  sum  Ua  +  36— 7c  may  be  ^^Titten  down  at  once. 

ORAL   EXERCISES 

Pick  out  similar  terms  and  add  mentally : 

1.  ^x  +  3y  —  2z\   —2x  —  7y-\-4:Z. 

2.  —  3  ??i  +  7  ?i  —  6p  ;  6  m  —  T)  n -\-  3 p. 

3.  3  A:  —  7  r  —  5  8 ;  9  >•  —  7  A-  +  8  s. 

4.  3  b  +7  ax  +  2(1',  7  6  —  4  ax  -f  5  d. 

5.  5ax  +  2  bd  —  3  C]   —  7  a.T  —  bd-\-6  c. 

6.  3  m?i  +  4pg  —  7  rs ;    —  8  j)7  +  9  ?'s  —  0  mn. 

7.  3(a  +  6)  +  7(.v-?/);    _  9(a  +  6)  +  ll(.r  -  ;y). 

8.  3x-2y-{-4.(a-b);   -  S  x -^  6  y -\-S(a  -  b). 

9.  4a;  —  2?/  +  32;;   — 5  a; +  4^— 82;  7  x  -{-3y  —  Sz. 


72  ALGEBRAIC   EXPRESSIONS 

60.  Subtraction  of  Polynomials.  Since  subtraction  is  per- 
formed by  adding  the  subtrahend,  with  its  sign  changed,  to  the 
minuend,  we  arrange  the  terms  as  in  addition. 

This  is  illustrated  as  follows : 

From  15  ah  —  nxy-\-  11  rt  subtract  —  5  a6  -f-  4  xy  —  5  rt. 

Arranging  as  on  page  69  and  applying  Principles  I  and  VIII : 

15  a&  —  VI  xtj  +  11  rt 

—  5  ab  +    4  xy  —    5  rt 

20  ab  -2lxy  +  16  rt 

As  suggested  in  §  45,  it  is  suflBcient  to  change  the  signs  of  the  sub- 
trahend mentally,  rather  than  to  rewrite  them  before  adding  to  the 
minuend. 

ORAL  EXERCISES 

Pick  out  the  similar  terms  and  subtract  mentally. 

1.  From  5x  —  3y  -{-7  z  subtract  2  x-\-7  y  —  9  z. 

2.  From  7  r  —  7  s  +  5  ^  subtract  9  r  +  8  ^  —  4  s. 

3.  From  6  m  —  9  n  —  ]?  subtract  —  4  m  -\-  12  n  —  S  p. 

4.  From  o  ab  —7  be  -\- 11  ac  subtract  12ab  —4:bc  —  4  ac. 

5.  From2(a;-3) +3(.?/-4) -f  4(^ -5) 

subtract  -  4(ic  -  3)  +  2(y  -  4)  +  o{z  -  5). 

6.  From  5a  —  b-{-2c— 7(x~  y) 

subtract  —  11  a  —  86  —  5c+  4(.«  —  y). 

7.  From  2  xy  —  3{x  —  y)-\-  byz 

subtract  —  S  xy  —  7{x  —  y)  —  6  yz 

8.  From  6  ay -3bz— S  (a  -  b) 

subtract  9(a  —  b)  -\-  G  bz  —  11  ay. 

9.  From  2(a  +  b  -\- c)-}- 2  x  -  3  y 

subtract  6(a  +h  -\-  c)  —3x-\-  5  y. 

10.  Fvom  12(a -^b -c)-3  ab -\-2  a- 3  b 

subtract  —  3  (a  -f  6  —  c)  4-  6  ab  —  5  a  -\-  2  b. 

11.  Ywm5x-7  a-\-Sb  -\-3c-9d 

subtract  —  3  6  -|-  G  c  —  0  (i  —  3  .c  +  3  a. 


SUBTRACTION   OF   POLYNOMIALS  73 

WRITTEN    EXERCISES 

Arrange  similar  terms  in  columns. 

1.  From  9  X  -{-  o  1/  —  11  z  subtract  —  5  x  -{-  S  y  —  3  z. 

2.  From  12  ab  —  3  ccl  +  12  xy  subtract  3ab  -{-2  cd  —  11  xy. 

3.  From  9  xc  +  ^ad  —  3cz  -\-6  y  subtract  3  y  —  3  ad  -\-  o  cz. 

4.  From  13  t  -\-  o  mx  —  5  cv  subtract  21  —  1  mx  —  3  cv. 

5.  From  3  v  —  2  iv  +  5  mn  —  4  xz  subtract  —  v-\-5  io  —  3  mn. 

6.  From  311)  -\- 4:  xy  -\-  IG  ax  —4  subtract  ^h  —  d  xy  —  3  ax. 

7.  From  1—3  a—  o  xz  —  3  vy  —  x  subtract  1  a-\-2  xz-\- 1  vy. 

8.  From  ^xy  —  3  x-^  ly  subtract  —2xy-\-13iv-{-lx—  2  y. 

9.  From  2  a6  —  5  +  7  v  -\- 13  abc  subtract  3  a6  +  v  +  8  ahc. 

10.  From  8  acx—  4  by  — 3  cy  subtract  4  acx  4-  2  by-{-  4  cy-ld. 

11.  From  31  '  15  —  7  xy  subtract  12  -45-1-9  xy. 

12.  From  3  abc  —  1  -\-  2{x  -\- y)  —  3  xy 

subtract  28  -f-  4  xy  —  3(x -\-y)-\- S  abc. 

13.  From  21  +  9{xy  -  z)  -\-  3(a  -f-  b) 

subtract  S(xy  —  z)  —  8(a  -\-  b)  -\-  15. 

14.  From  5ax—3by-\-4:ax-^D  by  subtract  5  6^-|-  3  ax-\-l  by. 

15.  From  15-48+8  ab  -f  49  x  subtract  7-48-9  ab  -14  x. 

16.  From  19(r  -  5  s)  +  13(5  x  -  4)  +  7{x  -  y) 

subtract  17(5  x  —  4)  —  d{x  —  ?/)  —  ll(r  —  5  s). 

17.  From  30  +  1  l(x  -  o  yz)  -  13(5  y  -  z) 

subtract  32  -\- S(5  y  —  z)  —  7 (a;  —  5  yz). 

18.  From  a{b  +  c)  +  4(?7i  +  n)  —  16  c 

subtract  9(m  +  ?<)  +  31  c  —  d{b  +  c) . 

19.  From  5(7  x  -  4)  +  3(5  ?/  -  3  .r)  +  35 

subtract  56  —  9(7  ic  —  4)  +  8(5  y  —  3  x). 

20.  From  (3  a  +  9  6  -  12  c)  -f-  3 

subtract  (6  a  -  12  6  -  18  c)  --  6 

21.  From  (axi/  +  ayz  —  axz)  -r-  a  subtract  y(x  +  2)  —  2  xz. 

22.  From  (abc  +  cxy)  -=-  c  subtract  2  a6  —  2  xy. 


74  ALGEBRAIC   EXPRESSIONS 

EXERCISES   IN   ADDITION  AND   SUBTRACTION 

1.  Add  5  X—  3y-7  r-^St,  —7x  +18  y —  4:r,  —7  t —  20x, 
-  24  2/  + 18  r  -  15 1,  and  13  x  +  15  2/  + 11  r  +  6  ^. 

Check  the  sum  by  substituting  x=l,  y  =  1,  ?•  =  1,  t  =  1. 

2.  Add  17  a  —  9  6,  3  c  4- 14  a,  b  —  3  a,  a  —  17  c,  a  —  S  b,  and 
-h  4  c.     Check  for  a  =  1,  6  =  2,  c  =  3. 

3.  Add  2x-\-3y  -t,   —6y  +  St,   —x  +  y  —  t,   —  4:t-\-7 x, 
and  3  y.     Check  by  putting  each  letter  equal  to  1. 

4.  Add   17  r  +  4  6"  -  ^,    2t-\-3u,   2  r  -  3  s  +  4  ^,   bu-Qt, 
7  r  —  3  s  -^'^11,  and  8  r  —  2  ?  -|-  6  u.     Check  as  in  Example  3. 

5.  Add  3  7i  -f  2  ^  +  4  ?^  and  h  +  3t^3u.    Check  by  putting 
h  =  100,  t  =  10,  n  =  l;  i.e.  324  +  133  =  457. 

6.  Add  4  /i  +  3  ^  +  u  and  3/^  +  2^  +  7?^.     Check  as  in  5. 

7.  Write  247,  323,  647,  239,  and  41,  as  number  expressions 
like  those  in  Exs.  5  and  6  and  then  add  theni. 

8.  Add  647,  391,  276,  and  444  as  in  Ex.  7. 

9.  Adidi4^  t— u,  bt  —  u,Q>t—u,7  t  —  u.    Check  for  ^=10,  t<=  1. 

10.  Simplify :  3  xyz  —  2  xyz  -f-  5  xyz  —  4  xyz  -f  xyz  —  xyz. 

11.  Subtract  5a  —  3  6  +  6c  from  —  8  a  +  7  6  —  11  c. 

12.  From  7  xy-\-^  xz  ■\- ^  yz  subtract  17  ic_y  —  19  xz  —  20  yz. 

13.  From  6  a  —  3  ?/  subtract  ^y  —  3z. 

14.  From  3^)  —  4g  +  8r  subtract  7  p  —  11  r  -]-  11  q. 

15.  From  2  X  —  3y  subtract  5  x  -{-7  y  -\-2  a  —3  b. 

16.  From  the  sum  of  18  abc  —  27  xyz  -\- 13  rst  and  —  11  ahc 
-h  1 6  xyz  —  52  rst  subtract  67  r^t  —  39  ahc. 

17.  To  the  difference  between    the  subtrahend  15x  — 18?/ 
+  27  2;  and  tlie  minuend  117  x  +  97  ?/  —  81  2;  add  4  .t  —  6  _?/  4-  3  z. 

18.  Add    ll(a;  —  ?/)  4- 15  (a  -  b)  and  -  20  {x  —  y)  —  37(a  -  b) 
and  from  the  sum  subtract  135  {x  ~  y)  —  213  (a  —  b). 


EXPRESSIONS   IN   PARENTHESES  75 

ALGEBRAIC   EXPRESSIONS   IN   PARENTHESES 
I 

61.  Removing  Parentheses.  The  sign  +  before  parentheses 
means  that  each  term  within  is  to  be  added  to  what  precedes, 
and  the  sign  —  means  that  each  term  within  is  to  be  sub- 
ti'acted  from  what  precedes. 

By  Principle  A^II 

a  -\-  (-\-  h)  =  a  -{-  b  and  a  -\-  (—  b)  =  a  —  b  ; 

and  by  Principle  VIII, 

a  —  (+  b)  =  a  —  b  and  a  —  (—  b)  =  a  +  b. 

Hence,  we  have 

Principle  XII 

62.  Rule.  ParentJieses  preceded  by  the  sign  +  may  be 
removed  without  further  change. 

Parentheses  preceded  by  tJve  sign  —  may  be  removed  by 
changing  the  sign  of  each  terin  within. 

iSTote  that  in  each  case  the  sign  preceding  the  parentheses  is 
also  removed  after  the  operation  indicated  by  it  has  been 
carried  out,  and  that  if  no  sign  is  written  before  the  first  term 
in  the  parentheses,  the  sign  +  is  understood. 

Kemove  the  parentheses  and  simplify  the  following : 

Example  1.     3  «  +  (a  -  /^  +  4)  -  (2  a  +  8  6  -  2) 

=  3a-f«-&+4-2a-36  +  2r=2a-46  +  f) 
Example  2.     5(3a:  +  ?/)  -  4(2  x  -  3  ?/ +  2) 

=  15a;  +  5?/-8a:+12//-8  =  7x4-17?/-8. 

In  Example  2  we  multiply  the  terms  within  the  first  parentheses  by  5  and 
those  in  the  second  by  4  and  then  remove  the  parentheses  by  Principle 
XII. 

ORAL  EXERCISES 

Remove  parentheses  in  the  following : 

1.  a  4-  (a;  —  ?/).  5.    x  +  (a  +  6  —  c). 

2.  a—{x-\-y).  6.    x-\-{a  —  b  —  c). 

3.  a  —  {x  —  y).  7.    x—{a  —  b  —  c). 

4.  —(—  x  —  y).  8.    x  —  (—a  —  b—c). 


76  ALGEBRAIC    EXPRESSIONS 

WRITTEN  EXERCISES 

Keniove  the  symbols  of  aggregation  and  simplify : 

1.  (3x-2y)-{4.x-\-3y-  2). 

2.  X  —  y  —  2  z  —  (3  X  -\-  2  7j  —  7  z). 

3.  3(a  +  5  H-  c)  -  2(a  -b  +  c). 

4.  8(5  X  -  y  +  2  ^)  -  11(3  x  +  y-  z). 

5.  5(7  a;  -  4  ?/)  +  9{x  -  y)  -  3(2  a:  +  3  2/). 

6.  8(r  _  s)  +  (2  r  +  s)  -  (?-  -  2  s). 

7.  11  ^  +  (2^-1) -(1-3^). 

8.  9(r-s)-3(r  +  .s)-+  2(2  r  -  s). 

9.  3(5.T-7  2/)-(4x-3.y  +  2)-5y. 

10.5  .r  -  (8  -  4  x  +  7  ?/)  +  (5  .t  +  3)  -  (5  ?/  +  3  a.-  -  99). 

11.  _  (3  a  _|-  5  6  -  7  c)  +  (8  a  -  4  c)-  (9  c  -  4  6  +  4  a)  -91  a. 

12.  7- (4-4  c  +  2  rt-2  a) +31  c- (4-2  a-5d.)- (-8  c). 

13.  (41a6-21c-f-4)-(36c  +  15-78a?>)  +  (13c-90a6-8). 

14.  9  %-(4  c-8  %-13)-2  c-16-(34  hy-12  c  +  ^  by). 

15.  C)  }nn-\-(  —  9  m  —  7  ?i  +  14)  — 8  n+(13  m)i  — 17  m)H-34m?i. 

16.  34  ax  -  (~  17  aa;  -f  42)  +  8  .T  -  (14  a  +  24  ax  -  7). 

17.  19 -(2-  7  a-4:b  -^  11  ab)-(- 2  b -^S  ab  +  4  a). 

18.  11  by  -(Ab  -IS  y  +  17  by)- {-  5  b  -17  by  -\- 13  y). 

19.  39  rs  -  20  s-19  r  -(7  rs  +  8  s-  19  ?')-(15  r  -  5s  -  50). 

20.  x-\Sx-(2y-3x)  +  (2x-'ly)\. 

Suggestion.     First  remove  the  parentheses,  then  the  braces  or  brackets. 

21.  a  +[a  —  (b  +  c)— 2c]. 

22.  a-\-(a-b)-\-(3a-2b)\. 

23.  2a;-3(^;-l)-[a;-2(2.i'-l)]. 

24.  a  —  \a+(b  —  c)—2(a-\-b  +  c)\. 

25.  -  ]a-la-\-(b  —  c)-2(a  +  b-\-c)']l. 

26.  -  \  —[—(-a  —  b-\-c-d)^\. 

27.  -  i  +l-(-a-b-c  +  d)^l. 


EXPRESSIONS  IN   PARENTHESES  77 

63.  Removing  all  Symbols  in  Order.     In  such  expressions  as 

I  11       III  IV 

2a  —  J4(a  —  b)  —  ('3a  —  2b)l,  the  symbols  of  aggregation  may 
be  removed  in  order  as  we  read,  by  noting  that  a  term  affected 
by  an  even  numbev  of  minus  signs  is  plus,  while  one  affected 
by  an  odd  number  is  minus. 

Thus,  in  the  above  expression,  4  a  is  affected  by  one  minus  sign  (I),  4  ?) 
by  two  (I  and  II),  3  a  by  two  (I  and  III),  and  2  6  by  three  (I,  III,  and 
IV) .     Hence  the  expression  equals  2a  —  4a  +  4&4-3a  —  26  =  a  +  2  6, 

ORAL  EXERCISES 

Kemove  all  symbols  of  aggregation  as  you  read,  then  write 
the  result,  if  necessary,  and  simplify. 

1.  rt-(2a  +  3^).  6.  a  +  l2a-{-{Sa-b)l. 

2.  x-[2x-{x-{-y)^.  7.  a -^{2  a -(3  a  -  b)}. 

3.  7?i  —  [3(m  — 7i)— 4  m].  8.  a  —  \2  a-\-{3  a  —  b)]. 

4.  r  +  (Sa-2b)-{a-b).  9.  a  -  )2  j -(3  a -f  5)(. 

5.  a-{2a-{3a-b)\.  10.  «- |2  a-[3  a-(a  +  6)]  J- 

11.  Read  Examples  20  to  27  in  the  preceding  exercises  in 
this  manner. 

64.  Inserting  Expressions  in  Parentheses.  By  the  converse 
of  Principle  XII  terms  may  be  inclosed  in  parentheses  with  or 
without  change  of  sign,  according  as  the  sign  —  or  +  precedes. 

JS.g.  a-\-b  —  c  =  a+(b  —  c)  and  a  —  b  +  c  =  a  —  (b  —  c). 

ORAL   EXERCISES 

In  each  of  the  following,  place  the  last  two  terms  in 
parentheses. 


1. 

X  —  y  +  2. 

7. 

5  xy  —  0  X  -{-  3  y. 

2. 

5  a  +  3  6  -  c. 

8. 

9  ax  -\- 3  by  —  4:  cd. 

3. 

m  —  n-\-  2). 

9. 

a  —  3b-\-d  —  5c. 

4. 

5a-36  +  2c. 

10. 

13  -  7  a  -  3  6. 

5. 

7m  —  4:71  —  3  p. 

11. 

19  .T  -  3  c  +  4  e. 

6. 

8  +  4  6  -  3  c. 

12. 

21  ax  -  13  bx  H-  6  dx. 

78  ALGEBRAIC   EXPRESSIONS 

WRITTEN  EXERCISES 

In  the  following,  use  either  method :  (1)  remove  one  symbol 
at  a  time,  beginning  with  the  innermost ;  (2)  remove  them  in 
order  as  you  write,  beginning  at  the  left. 


1.  —  [o  +  5  a  —  (a  —  x)  —  (a  —  X  —  a)  5  —  a] . 

2.  Sx-l6y-[3x-(2y-x)-Sy']  +  x\. 

3.  —[x  —  \z  -\- (x  —  z)  —  (z  —  x)  —  z\  —  x']. 

4.  2a-[2a-l2a~(2a-2a-a)\]. 

5.  —[_a  —  ] a  —  {x  —  a  —  X  —  a)  —  al  —  2  a^, 

6.  -l5x-\4.y-{oz  +  2y)-(2x-5z)\]. 

7.  iC)-x~l7x-\Sx-{9x-3x-6  x)l]. 

8.  2x-l3y-\4:X-(5y-6x-7  y)i']. 

9.  2a-[3  6+(2  6-c)-4c  +  52a-(3  5-c-26)n. 


10.  a  -[5b  -  la-(p  c  -  2  c  -  b  -  4.b)-{-  2  a  -(a  -  2b  -^c)\]. 

11.  2(3b-5a)-7[a-6\2-5(a-b)i]. 

12.  -2Ja-6[a-(5-c)]S-f  6J5-(c  +  a)j. 

13.  _3S-2[-4(-a)]S  +  5)-2[-2(-a)]S. 

14.  _25-[-(^-2/)](  +  S-2[-(aj-y)]S. 

15.  a  — (6  — c)-[a  — 6  — c  — 2J&  +c  — 3(c  — a)— fZj]. 

16.  2  X  -(3  y  -  4.  z)-  \2  X  -13  y  +  4:  z  -  3  y  -{4.  z  +  2  o^)]  j. 

17.  -  2(a-  d)-  2[6  -^c  +  d-3\c  +  cl-  4(f?  -  a) J]- 

18.  _4(a  +  c«)+4(6-c)-2[c+ri  +  a-3Jd+a-4(6  +  c)n. 

19.  a-2  b-[4:  a-6b-  \3  a-c-\-(5  a-2  b-3  a-c-\-2  b)\y 

20.  a-[-6J-c(-(^  +  e-/)  +  2a-cj  +  c  +  f?]. 

In  each  of  the  following  insert  the  last  three  terms  in  paren- 
theses j)receded  by  the  minus  sign : 

21.  rj-\-2a-3-^b.  26.  3b-4.-2x-^y. 

22.  x-4:-2a-\-c.  27.  9  ?/-3  c  +  8  -  2  a. 

23.  7xy-3x-2y  +  z.  28.  12  +  3rt-86  +  c. 

24.  a-2&-4c  +  (/.  29.  c-Sb- 3  d-^  12. 

25.  6  —  a  —  6  —  c.  30.  3  a  —  4  c  +  5  d  —  2  6. 


FORMING   ALGEBRAIC   EXPRESSIONS  79 

FORMING   ALGEBRAIC  EXPRESSIONS 
ORAL   EXERCISES 

1.  The  sum  of  two  numbers  is  20.  If  x  is  one  of  thera, 
how  may  the  other  be  represented  ? 

2.  The  sum  of  two  numbers  is  16.  What  is  a  convenient 
representation  of  each  number  ?      • 

3.  How  do  you  represent  six  times  the  number  x? 

4.  How  do  you  represent  6  less  than  twice  the  number  n  ? 

5.  How  do  you  represent  8  more  than  4  times  the  number  a  ? 

6.  If  X  is  a  number,  how  do  you  represent  a  number  which 
is  8  more  than  i  of  this  number  ? 

7.  If  a  is  a  number,  how  do  you  represent  a  number  10 
less  than  ^  of  this  number  ? 

8.  A  number  is  represented  by  x,  another  number  is  15 
less  than  twice  this  number.  How  do  you  represent  the  sum 
of  these  numbers  ? 

9.  A  number  is  represented  by  n.  Another  number  is  6 
more  than  three  times  this  number.  How  do  you  represent 
twice  the  second  number  ? 

10.  A  number  is  represented  by  x.  Another  number  is  24 
less  than  four  times  this  number.  How  do  you  represent  i 
of  the  second  number  ? 

11.  A  number  is  represented  by  a.  By  how  much  does  this 
number  exceed  50  ? 

12.  The  difference  between  two  numbers  is  10.  How  may 
we  represent  the  numbers  ? 

13.  If  a  number  exceeds  another  by  50,  represent  twice  the 
smaller  number  plus  three  times  the  greater. 

14.  If  n  represents  the  number  of  years  in  my  age  now,  how 
old  was  I  five  years  ago  ?     How  old  will  I  be  10  years  hence  ? 


80  ALGEBRAIC   EXPRESSIONS 

PROBLEMS  ON  THE  RELATIONS   DF   NUMBERS 

1.  The  sum  of  two  numbers  is  20  and  one  is  4  greater  than 
the  other.     Find  the  numbers. 

2.  If  10  is  added  to  4  times  a  number,  the  result  is  74. 
Find  the  number. 

3.  Six  less  than  twice  3,  certain  number  equals  18.     Find 
the  number. 

4.  Eight  more  than  4  times  a  number  equals  48.     Find  the 
number. 

6.    Ten  more  than  ^  oi  a,  certain  number  equals  18.     Find 
the  number. 

6.  One  number  is  12  greater  than  another.     Their  sum  is  24. 
Find  the  numbers. 

7.  One  number  is  10  less  than  twice  another  number.     The 
sum  of  the  numbers  is  50.     Find  the  numbers. 

8.  One  number  is  18  more  than  3  times  another  number. 
The  sum  of  the  numbers  is  66.     Find  the  numbers. 

9.  Find  two  numbers  such  that  one  is  6  more  than  twice 
the  other,  and  the  smaller  plus  twice  the  greater  equals  42. 

10.  One  number  exceeds  another  by  50.     Twice  the  smaller 
number  plus  the  larger  number  equals  95.     Find  the  numbers. 

11.  The  difference  between  two  numbers  is  10.     The  smaller 
number  plus  twice  the  larger  equals  35.     Find  the  numbers. 

12.  One  number  exceeds  another  by  40.     Twice  the  smaller 
number  plus  3  times  the  greater  equals  295.     Find  the  numbers. 

13.  One  number  is  16  loss  than  6  times  another.      The  sum 
of  the  numbers  is  40.     Find  the  numbers. 

14.  The  sum  of  my  ages  10  years  henco   ind  5  years  ago  is 
55.     How  old  am  I  ? 


FORMING  ALGEBRAIC   EXPRESSIONS  81 

PROBLEMS   ON   THE   ARRANGEMENT   AND   VALUE   OF  DIGITS 

If  we  speak  of  the  number  whose  3  digits,  in  order  from 
left  to  right,  are  5,  3,  and  8,  we  mean  538=500  +  30-1-8. 
Likewise,  the  number  whose  three  digits  are  h,  t,  and  a  is  writ- 
ten 100  h  +  10  f  +  ?fc.     Note  that  htu  would  mean  h  x  t  x  u. 

Hence,  when  letters  stand  for  the  digits  of  a  number  written 
in  the  decimal  notation,  care  must  be  taken  to  multiply  each 
letter  by  10, 100, 1000,  etc.,  according  to  the  position  it  occupies. 

ORAL  EXERCISES 

1.  If  the  sum  of  two  digits  is  9  and  if  x  represents  one  of 
them,  how  do  you  represent  the  other  ? 

2.  If  the  difference  between  two  digits  is  3,  and  if  x  repre- 
sents one  of  the  digits,  how  do  you  represent  the  other? 

3.  Give  an  expression  representing  a  number  if  the  digit 
in  tens'  place  is  x  and  the  digit  in  units'  place  is  y.  Also  give 
an  expression  representing  the  number  if  the  order  of  the 
digits  is  reversed. 

4.  Give  an  expression  representing  a  number  if  the  digit 
in  tens'  place  is  x  and  in  units'  place  6  —  x.  Also  give  an  ex- 
pression representing  the  number  with  the  order  of  the  digits 
reversed. 

5.  Give  an  expression  representing  a  number  if  the  digit 
in  tens'  place  is  t  and  that  in  units'  place  t  -\-  6.  Also  give  an 
expression  representing  the  number  obtained  by  reversing  the 
order  of  the  digits. 

6.  If  the  digit  in  tens'  place  is  x  and  in  units'  place  12  —  x, 
give  an  expression  representing  7  times  the  sum  of  the  digits. 

7.  If  the  digit  in  tens'  place  is  x  and  the  digit  in  units' 
place  is  12  less  than  5  times  a;,  write  an  expression  representing 
the  number.     Also  express  it  with  the  digits  reversed. 


82  ALGEBRAIC   EXPRESSIONS 

Illustrative  Problem.  A  number  is  composed  of  two  digits 
whose  sum  is  6.  If  the  order  of  the  digits  is  reversed,  we 
obtain  a  number  which  is  18  greater  than  the  first  number. 
What  is  the  number  ? 

Solution.     Let  x  =  the  digit  in  tens'  place. 

Then,  Q  —  x  =  the  digit  in  units'  place. 

Hence,  the  number  is  10  a; +  6  — a;.  Reversing  the  order  of  the 
digits,  we  have  as  the  new  number  10(6  — 5c)+x. 

Hence,  10(6  -  a:)+ x  =  18  +  lOx  +  6  -  a:. 

Solving,  X  =  2,  the  digit  in  ten's  place, 

and  6  —  a;  =  4,  the  digit  in  unit's  place. 

Hence,  the  required  number  is  24. 

WRITTEN  EXERCISES 

In  each  of  the  examples  1  to  8  below  the  number  considered 
is  composed  of  two  digits. 

1.  The  digit  in  units'  place  is  2  greater  than  the  digit  in 
tens'  place.  If  4  is  added  to  the  number,  the  result  is  then  equal 
to  5  times  the  sum  of  the  digits.     What  is  the  number  ? 

2.  The  digit  in  tens'  place  is  3  greater  than  the  digit  in 
units'  place.  The  number  is  1  more  than  8  times  the  sum  of 
the  digits.     What  is  the  number  ? 

3.  The  sum  of  the  digits  is  9.  If  the  order  of  the  digits 
is  reversed,  we  obtain  a  number  which  is  equal  to  12  times 
the  remainder  when  the  units'  digit  is  taken  from  the  tens' 
digit.     What  is  the  number  ? 

4.  The  sum  of  the  digits  is  12.  If  the  order  of  digits  is 
reversed,  the  number  is  increased  by  18.     Find  the  number. 

5.  The  tens'  digit  is  2  less  than  its  units'  digit.  The  number 
is  1  less  than  5  times  the  sum  of  its  digits.    What  is  the  number  ? 

6.  The  digit  in  units'  place  is  4  less  than  that  in  tens' 
place.  If  the  order  of  the  digits  is  reversed,  we  obtain  a 
number  whi(;h  is  3  less  than  4  times  the  sum  of  the  digits. 
What  is  the  number  ? 


REVIEW   QUESTIONS  83 

7.  The  digit  in  units'  place  is  2  less  than  twice  the  digit  in 
tens'  place.  If  the  order  of  the  digits  is  reversed,  the  number 
is  unchanged.     What  is  the  number  ? 

8.  The  digit  in  tens'  place  is  12  less  than  5  times  the  digit 
in  units'  plac^.  If  the  order  of  the  digits  is  reversed,  the 
number  is  equal  to  4  times  the  sum  of  the  digits.  What  is 
the  number  ? 

9.  A  number  is  composed  of  three  digits.  The  digit  in 
units'  place  is  3  greater  than  the  digit  in  tens'  place,  which 
in  turn  is  2  greater  than  the  digit  in  hundreds'  place.  The 
number  is  equal  to  96  plus  4  times  the  sum  of  the  digits. 
What  is  the  number  ? 

REVIEW   QUESTIONS 

1.  What  is  a  polynomial  ?  a  term  .^  How  are  polynomials 
classified  ?  What  are  similar  terms  9  By  what  principle  are 
similar  terms  added?     By  what  principle  are  they  subtracted? 

2.  In  adding  or  subtracting  polynomials  how  may  the 
terms  be  arranged  for  convenience?  State  the  principle  on 
which  this  is  based. 

3.  AVhat  is  the  rule  for  removing  parentheses  when  preceded 
by  the  sign  +  ?  By  the  sign  —  ?  How  may  Principle  XII 
be  used  for  inclosing  terms  within  parentheses? 

4.  Tell  how  to  remove  all  the  symbols  of  aggregation  in 
order  as  you  read  ^  x— \x— {^x-\-y)  — (2  x  —  3y)\.  How 
many  minus  signs  affect  the  term  3  a??  How  many  affect  3  ?/? 
What  are  the  final  signs  of  these  terms? 

5.  Add  Principles  XI  and  XII  to  your  list  expressed  in 
symbols : 

XI,  a-\-b-\-c  =  a-\-c-{-b  =  c-\-a-\-b,  etc. 

XII.  a-^{b-c-d)  =  a-^b-c-d, 
a—  (b— c  —  d)  =  a  —  b-\-c-\-d 


CHAPTER   V 

MULTIPLICATION  AND  DIVISION  OF  ALGEBRAIC 
EXPRESSIONS 

MULTIPLICATION   OF   POLYNOMIALS 

In  multiplying  one  monomial  by  another  we  use 

Principle  XIII 

65.  Rule.  To  obtain  the  product  of  two  or  more  factors, 
these  jnay  he  arranged  and  multiplied  ijv  any  order. 

The  truth  of  this  principle  is  clear  from  examples  such  as  : 

2  .  3  .  5  =  2  .  5  .  3  =  5  .  3  .  2  =  5  .  (3  .  2)  =  2  .  (3  .  5)  =  30. 

HISTORICAL  NOTE 

Associative  and  Commutative  Laws  of  Factors.  Principle  XIII,  like 
Principle  XI,  states  two  fundamental  laws  of  algebra  :  (1)  the  associative 
Ihw  ot  tactors,  first  so  called  by  F.  S.  Servois  ;  (2)  the  commutative  law 
of  factors,  first  so  called  by  Sir  William  Hamilton. 

The  associative  law  of  factors  states  that  factors  may  be  grouped  in 
any  combination. 

Thus,  ahc  =  a(bc)  =  (ab)c. 

The  commutative  law  of  factors  states  that  factors  may  be  multiplied 
'Ogether  in  any  order. 

Thus,  abc  =  acb  =  cba,  etc. 

66.  Repeated  Factors.  In  multiplying  algebraic  expressions, 
the  same  factor  may  occur  more  than  once  in  the  same  term. 

Thus  we  may  have  5  >  5  ov  x  -  x.  These  are  written  o^  and 
x^  respectively,  and  are  read  5  square  and  x  square. 

This  is  a  convenient  way  of  abbreviating  written  expressions. 

E.g.  5  a  •  3  a  =  (6  •  3)  •  (a  •  a)  =  16  a'-^ ;  ay  •  ay  =  aayy  =  cfiy"^. 

84 


MULTIPLICATION  OF  POLYNOMIALS 


85 


67.  The  product  of  two  binomials  such  as  5  -f-  8  and  5  -\-3  may 
be  obtained  in  two  ways : 

(1)  (5  +  3)  (5  +  8)  =  8  .  13  =  104. 

(2)  (5  +  3)(5  +  8)  =  5(5  +  8)+  3(5  +  8)  =  52  +5  •  8  +  3  •  5+3  •  8  =  104. 

The  second  method  is  illustrated  by  the  accompanying  fig- 
ure in  which  5  -|-  8  is  the  length  of  a  rectangle  and  5  +  3  is 
its  width.     The  total  area  is  the  product  (5  +  3)  •  (5  +  8)  and 
it  is  composed  of  the  four  small  areas,  5^, 
5-8,  3-8,  and  3.5. 

The  second  method  here  used  for  mul- 
tiplying (5 -f- 3)  (5 -f  8)  is  the  only  one 
available  when  the  terms  of  the  binomials 
cannot  be  combined. 

Thus,      (x  +  4)  (x  +  6)  =  a;(x  +  6)  +  4  (a:  +6)  =  x2  +  6  X  +  4  a:  +  4  .  6 
=  x^-\-10x  +  24, 
and  (a  +  b)(m  +  n)  =  a{m  +  n)  +  6(m  +  w)=  am  +  an  +  hm-\-hn. 

Hence,  to  multiply  two  binomials,  multiply  each  term  of  one 
by  every  term  of  the  other  and  add  the  products. 

68.  Product  of  Two  Trinomials.     In  a  manner  similar  to  that 
just  illustrated  we  may  multiply  two  trinomials. 


3-5 

3-8 

5-5 

5 

5-8 

8 

am 

an 

ar 

bm 

bn 

hr 

cm 

en 

cr 

From  the  figure  we  see  at  once  that 

(a  +  6  +  c){m  +  M  +  r)  =  am  +  bm  +  cm  +  an  +  bn  -\- en  +  ar  +  br  +  cr, 

in  which  each  term  of  one  trinomial  is  multiplied  by  every  term  of  the 
other  and  the  products  are  added. 

Evidently  the  same  process  is  applicable  to  the  product  of 
two  such  polynomials  each  containing  any  number  of  terms. 


86  ALGEBRAIC   EXPRESSIONS 

WRITTEN  EXERCISES 

Find  the  following  products  : 

1.  (x-\-l)(x-^2).  14.  (5s  +  l)(s  +  5). 

2.  (x-hS)(x4-5).  15.  {x  +  7){3x  +  4:). 

3.  (u-{-7)(u-\-4r).  16  (a  +  4)  (3  a -f- 1). 

4.  (a +  8)  (a +  8).  17.  {3 -\-x)(2 -^5x). 

5.  (^+3)  (^  +  7  .  18.  (a  +  6)(3rt4-7  6). 

6.  Cv  +  9)  (2/ +  2).  19.  (a^  +  2/)(2a.-  +  32/). 

7.  (a;  +  1)  (a? -h  7).  20.  (7a.'H- 4)(x -f  8). 

8.  (s  +  5)(s  +  3).  21.  (a? +3)  (2  a;  4- 3). 

9.  (ci  +  ft)(c  +  d).  22.  (2a  +  5)(a  +  7). 

10.  (x  +  4)  (a;  +  3).  23.  (8  6  +  3)  (2  6  +  3). 

11.  (x  +  y)  (a  +  6).  24.  (5  a  +  4)  (2  a  +  3). 

12.  (2  a;  +  3)  (a?  +  2).  25.  (9  +  2  a?)  (5  +  a;). 

13.  (5  +  x)(6-\-  x).  26.  (6  +  2/)  (3  +  4  y). 

Example.     Solve  the  equation 

(a;  +  l){x  +  2)=  x{l  +  a;)  +  12.  (1) 

Solution.     Performing  the  indicated  multiplication, 

x:^  +  Sx  +  2=x  +  x^+12  (2) 
Subtracting  x^  from  both  sides, 

3x+2  =  x+12.  (3) 

By  Six,  2,  2  x=  10.  (4) 

By  i>  1  2,  a;  =  5.  (5) 

Equation  (2)  differs  from  those  solved  heretofore  in  that  it  contains 
the  term  x^  in  each  member. 

We  may,  however,  subtract  x^  from  each  member,  giving  equation  (3), 
which  is  a  form  already  studied. 

Solve  the  following  equations  : 

27.  (x-{-3){x-^2)  =  {x-\-l){x  +  2)-\-(5. 

28.  (a;-i-2)(a;  +  4)  =  (x+3)(a;H-l)  +  9. 

29.  (x  4-  3)(a;  -f-  5)  -  13  =  («  +  2)(a?  +  4). 

30.  {x  +  l){x  +  7)  -  2  =  (a;  -h  2)(a;  +  5). 


MULTIPLICATION   OF  POLYNOMIALS  87 

WRITTEN  PROBLEMS 

1.  The  length  of  a  rectangle  is  6  feet  greater  than  its  width 
10.     Express  the  area  of  the  rectangle  in  terms  of  w. 

2.  The  width  lo  of  a  rectangle  is  10  feet  less  than  its  length. 
Express  the  area  of  the  rectangle  in  terms  of  w. 

3.  The  length  of  a  rectangle  is  4  greater  than  its  width  w. 
Express  the  dimensions  of  this  rectangle  in  terms  of  w  after 
the  width  is  increased  by  2  and  the  length  by  3. 

4.  The  width  w  of  a  rectangle  is  8  less  than  its  length. 
Express  the  dimensions  of  this  rectangle  in  terms  of  w  after 
its  length  is  increased  by  2  and  its  width  is  increased  by  4. 
Also  express  the  area  of  the  new  rectangle  in  terms  of  iv. 

5.  A  field  is  10  rods  longer  than  it  is  wide.  If  its  length 
is  increased  by  10  rods  and  its  width  increased  by  5  rods,  the 
area  is  increased  by  640  square  rods.  What  are  the  dimen- 
sions of  the  field  ? 

Suggestions.     Let         ic  =  the  width  of  the  field. 
Then,  to  +  10  =  its  length, 

and  w(w  -\-  10)  =  its  area. 

The  area  of  the  increased  field  is  (w  -\-  5){io  +  20). 
By  the  conditions  of  the  problem, 

l^w  +  5)(m7  +  20)  =  w{io  +  10)  +  640. 

6.  A  rectangle  is  7  feet  longer  than  it  is  wide.  If  its  length 
is  increased  by  3  feet  and  its  width  is  increased  by  2  feet,  its 
area  is  increased  by  60  square  feet.    What  are  its  dimensions  ? 

7.  A  farmer  has  a  plan  for  a  granary  which  is  to  be  12  feet 
longer  than  wide.  He  finds  that  if  the  length  is  increased  8 
feet  and  the  width  is  increased  2  feet,  the  floor  space  will  be 
increased  by  160  square  feet.     What  are  the  dimensions? 

8.  If  the  length  of  a  rectangular  flower  bed  is  increased 
3  feet  and  its  width  is  increased  1  foot,  its  area  will  be  increased 
by  19  square  feet.  What  are  its  present  dimensions,  if  its 
length  is  4  feet  greater  thnn  its  width  ? 


88 


ALGEBRAIC   EXPRESSIONS 


69.    Multiplying  Polynomials  with  Negative  Terms.     In  §§  67 

and  68  we  studied  the  multiplication  of  polynomials  whose 
terms  were  all  positive.  The  same  method  may  be  applied  to 
polynomials  having  negative  terms. 

Example.  Find  the  product  of  (5  -  2)  and  (4  —  3).  This 
product,  written  out  term  by  term  as  a  product  of  two  sums, 
would  give 

(5  -  2)(4  -  3)  =  [5  +(-  2)][4  +(-  3)] 

=  5  .  4  +  5(-  3) -f  4(-  2)  +  (-  3) (-  2) 
zz:  20  -  15  -  8  +  6  =  3. 
Also  (5  -  2)  (4  -  3)  =  3  .  1  =  3. 

Similarly,      (x  +  6)(x  -  2)  =  x'^  -{-  6x  —  2 x -  10  =  x^  +  Sx  —  10, 
and  (x  —  3)  (x  —  5)  =  x^  —  3  X  —  5  a;  +  15  =  x^  —  8  a;  +  15. 


WRITTEN  EXERCISES 

Perform  the  following  indicated  operations  : 


1. 

{x-5)(x-3). 

18.    C 

2. 

[X-3)(X+4:). 

19.    ( 

3. 

(a-6)(a-l). 

20.    (( 

4. 

(u  +  5)(u-3). 

21.    ( 

5. 

(6  +  2)(6-7). 

22.    ( 

6. 

(3-6)(4  +  6). 

23.    (, 

7. 

(3  +  ^)(7-3aj). 

24.   (: 

8. 

(n-4)(3-n). 

25.    (< 

9. 

{a-b){c-^d). 

26.    (. 

10. 

{a-b){c-d). 

27.    C 

11. 

(a;-4)(x-5). 

28.    (( 

12.    ( 

[a-\-b  —  c){m  —  n). 

29.    ( 

13. 

{a-h)(l  a-\-3h). 

30.    (« 

14. 

(5-2/)(5a?  +  32/). 

31.    G 

15. 

(2a-36  +  c)(m  +  7i). 

32.    (, 

16. 

{v-t){lv-bf). 

33.    0 

17.    1 

^3a-2)(26-3). 

34.    (■ 

6  +  3a-6)(4c-9d-l> 
a  -\-  711  +  n){x  —  y  -\-  z). 
a+b-c)(d~e-\-f). 
V  -]-t  -\-u)(v  —  t  —  u). 
3x-5)(2x-\-7). 
2x-y-l){x-\-y). 

a-Sb-l){2a-b). 
2x-l  +  y)(Sx-2y). 
4a-6)(a  +  2  6). 
6x-\-5  y){2  X  -  y). 
1  m  —  3  n){2  m  -\-  5  n). 
8a-36)(2a-f  3  6). 
5c  +  rf)(2c-3d). 
3a-4  6)(3a  +  4  6). 
9  a;  — 4  i/)(4a;4-9?/). 
12a-5  6)(3a-6). 


MULTIPLICATION   OF   POLYNOMIALS 


89 


The  preceding  exercises  illustrate 

Principle  XIV 

70.  Rule.  The  product  of  two  polynomials  is  found  by 
multiplying  each  term  of  one  hy  every  term  of  the  other, 
and  adding  the  products. 

71.  It  should  be  observed  that  Principle  XIV  involves  a  re- 
peated application  of  Principle  II.     Thus 

(a  +  h){c+  d)  =  {a  +  b)c+(a+  h)d  =  ac -\- he  +  ad -^  hd. 

Arranging  the  Terms  of  the  Product.  In  multiplying  polyno- 
mials, the  work  is  usually  arranged  so  that  similar  terms  in  the 
product  are  written  in  columns  and  then  added. 

Example  1.     Multiply  'dx  —  2hj2x  —  ^. 

Solution.  3x  —  2 

•2x-b 
6  x"^  —  4  x 

-ISx-flO 
6  a;2  -  19  a;  +  10 

Example  2.     Multiply  3a;  —  22/-f2by4a;  —  3?/  —  2. 

Solution.  3  a:  —  2  y  -f  2 

4a;-3y-2 
12  a:2  -  8  a;?/  -f  8  a; 

—  6x  -h  4?/  —  4 


12  a:2  -  17  xi/  4-  2  X  -H  6  ?/2  -  2  y  -  4 


WRITTEN  EXERCISES 

In  each  case  simplify  the  expressions  within  the  parentheses 
as  much  as  possible  before  multiplying: 


1.  {x-7){3x  +  4:). 

2.  (x-2)(9a:  +  4). 

3.  {a  —  x){9x-\-4:a). 

4.  {nb+3a){2b-3b+o). 

5.  (:x-2-\-y)(4.y  ^3x). 


(a,_5x  +  4)(8?/-3-^v). 
(T-\-y-x){2y-\-x-l). 
(ox-{-3y-l){x-2). 
{x-y  +  3){5x-3y  +  5). 


10.    (a  —  13  n)(a  -  ?i  -f-  8). 


90  ALGEBRAIC   EXPRESSIONS 

WRITTEN   EXERCISES   AND   PROBLEMS 

Multiply  the  following : 

1.  (lSa-b-12a)(2b-3a). 

2.  (6  —  4  .T  +  3  x){7  x-{-y  —  8x-\-  1). 

3.  {ox-\-3y-4:X-2y){6y-\-Sx-2y-\-y). 

4.  (llb-a-10b)(6a-Sb-2a). 

5.  (— 7a  — l  +  8a)(5a  — 8  — 3a). 

Solve  the  following  equations  and  check  the  results : 

6.  (x-{-2){x-{-3)  =  {x-3){x  +  10)-{-10. 

7.  {5x-4.)(6-x)-97=(x-l)(6-5x). 

8.  (3  n  -  1)(18  -  7i)  =  {n  +  6)  (16  -  3  n). 

9.  (7-a)(9a-8)=31+(36-9a)(a  +  2). 

10.  (4  a  +  4)(a  -  3)  =  (4  a  +  l)(a  +  7)-  13  a  +  221. 

11.  (71  +  6)(3  n  -  4)-  14  =(n  +  8)  (3  n  -  3). 

12.  (8  n  +  6)(10  -n)+  150  =  (1  -  n)(8  7i  +  3). 

13.  (a  -  1)(13  -  6  a)  =  (6  a  -  3)(8  -  a)-  21. 

14.  (Tx-  13)(6  -x)-  (x  +  4)(3  -  7  a.-)=  70. 

In  the  following  make  sure  that  the  solutions  are  correct  by 
doing  the  work  with  care  and  looking  it  over  a  second  time. 

15.  (2x-3)(5x-^2)-7x  =  (2-5x)(7-2x)-^l. 

16.  (2  a  -  1)(3  a  -f  2)  -f-  a  =(4  -  3  a)(l  -  2  a)  +  7. 

17.  (Sx-l)(x-^7)-{-4:X  =  {2x-{-S)(4.x  +  6)-^5. 

18.  (5b  +  2)(3-b)  =  (3-5b)(A  +  b)-\-9. 

19.  (3  a -7)(2+a)  =  (5  +  3a)(2  4-a)• 
20.  {x-3){2x-\-5)-(3-x)(5-2x)=0. 

21.  (32/  -5)(5  -6y)-(9y-  6)(3-2y)=5. 

22.  (4  -  a)(3  -  a)  +  (a  -  2)(a  -  5)  +  (l  -  2  a)(2  +  a)  =  7. 

23.  Find   two   numbers   whose   difference  is  6  and  whose 
product  is  180  greater  than  the  square  of  the  smaller. 


MULTIPLICATION   OF   POLYNOMIALS  91 

24.  There  are  four  consecutive  even  integers  such  that  the 
product  of  the  first  and  second  is  40  less  than  the  product  of 
the  third  and  fourth.     What  are  the  numbers  ? 

25.  There  are  four  consecutive  integers  such  that  the  prod- 
uct of  the  first  and  third  is  223  less  than  the  product  of  the 
second  and  fourth.     What  are  the  numbers? 

26.  Find'  four  numbers  such  that  the  second  is  5  greater 
than  the  first,  the  third  5  greater  than  the  second,  and  the 
fourth  5  greater  than  the  third.  The  product  of  the  first  and 
second  is  250  less  than  the  product  of  the  third  and  fourth. 

27.  A  club  makes  an  ec^ual  assessment  on  its  members  each 
year  to  raise  a  certain  fixed  sum.  One  year  each  member  pays 
a  number  of  dollars  equal  to  the  number  of  members  of  the 
club  less  175.  The  following  year,  when  the  club  has  50  more 
members,  each  member  pays  $  5  less  than  the  preceding  year. 
What  was  the  membership  of  the  club  the  first  year  and  how 
much  did  each  pay  ? 

PROBLEMS   ON  RECTANGLES  AND   TRIANGLES 

28.  A  rectangle  is  10  inches  longer  than  wide.  Express  its 
area  in  terms  of  the  width  lo.  If  the  width  is  increased  by  4 
and  the  length  by  6  inches,  express  the  area  in  terms  of  iv. 

29.  A  rectangle  is  8  inches  longer  than  wide.  Express 
its  area  in  terms  of  the  width  iv,  after  the  width  is  increased 
4  inches  and  the  length  decreased  10  inches. 

30.  A  rectangle  is  5  feet  longer  than  it  is  wide.  If  it  were 
3  feet  wider  and  2  feet  shorter,  it  would  contain  15  square  feet 
more.     Find  the  dimensions  of  the  rectangle. 

31.  A  rectangle  is  6  feet  longer  and  4  feet  narrower  than  a 
square  of  equal  area.  Find  the  side  of  the  square  and  the 
sides  of  the  rectangle. 


92  ALGEBRAIC    EXPRESSIONS 

If  h  is  the  base  of  a  triangle,  h  its 
height  or  altitude,  and  a  its  area,  then 
area  =  \  {base  x  altitude) ; 

i.e.  a  =  ^bh. 

b 

32.  The  base  of  a  triangle  is  2  inches  less  than  its  altitude 
a.     Express  the  area  in  terms  of  a. 

33.  The  altitude  of  a  triangle  is  7  greater  than  its  base  h. 
If  the  altitude  is  increased  by  8  and  the  base  by  6,  express  its 
area  in  terms  of  h. 

34.  The  altitude  of  a  triangle  is  16  inches  less  than  the 
base.  If  the  altitude  is  increased  by  3  inches  and  the  base 
by  2  inches,  the  area  is  increased  by  52  square  inches.  Find 
the  base  and  altitude  of  the  triangle. 

PRODUCTS   OF  POWERS   OF  THE   SAME  BASE 

72.  Exponents;  Powers.  Any  number  written  over  and  to 
the  right  of  a  number  expression  is  called  an  index  or  exponent. 
If  an  exponent  is  a  x^ositive  integer,  it  shows  how  many  times 
the  expression  is  to  be  taken  as  a  factor. 

A  product  consisting  entirely  of  equal  factors  is  called  a 
power  of  the  repeated  factor.  The  repeated  factor  is  called  the 
base  of  the  power.     See  §  66. 

E.g.  x^  means  x  •  x  ■  x  and  is  read  the  third  power  of  x  ov  x  cube;  0(^ 
means  x  •  x  ■  x  -  x  •  x,  and  is  read  the  fifth  power  of  x  or  briefly  x  fifth. 
In  both  these  cases  the  base  is  x.  (x  —  y)^  =  (x  —  y)  (x  —  y)  (x  —  y)  and 
is  read  x  —  y  cubed  or  the  cube  of  the  binomial  x  —y. 

The  first  power  of  x  is  written  without  an  exponent.  Thus  x 
means  x^ ;  2  means  2^,  etc. 

Difference  between  a  Coefficient  and  an  Exponent.  A  coefficient 
i.H  a  factor,  while  an  ex})onent,  if  a  positive  integer,  shows  how 
many  times  some  number  is  used  as  a  factor. 

E.g.    5  a  =  5  •  a  which  means  a+a  +  a4-«  +  a,  while  a^  =  a  ■  a-a-  G'Q. 


PRODUCTS   OF  POWERS   OF   THE   SAME   BASE 


93 


ORAL   EXERCISES 

Find  the  following  powers  : 


1. 

23, 

2\    2\ 

7. 

92,        102. 

13. 

602,    702^ 

2. 

32, 

3'. 

8. 

112,       122. 

14. 

802,    902. 

3. 

42, 

43. 

9. 

132,       142. 

15. 

1002,    10002 

4. 

52, 

5'. 

10. 

152,    162. 

16. 

2002,  3002. 

5. 

62, 

61 

11. 

202,    302. 

17. 

4002,  5002. 

6. 

72, 

82. 

12. 

402,    502. 

18. 

6002,  7002. 

WRITTEN   EXERCISES 


Find  the  following  powers 


1. 

172, 

182. 

2. 

192, 

212. 

3. 

222, 

232. 

4. 

242, 

252. 

5. 

262, 

272. 

6. 

282, 

292. 

7. 

(a  +  h)\ 

13. 

{a-h-  cf. 

8. 

(C  -  d)2. 

14. 

(3  a  -  2)2. 

9. 

(a  +  6  +  c)2. 

15. 

{x-y  +  zf. 

10. 

(a  +  6  —  c)2. 

16. 

{2x-3yy. 

11. 

(3  -  of. 

17. 

(5  rt  -  2  by. 

12. 

{3-b-  c)\ 

18. 

(4.x  +  3yy, 

73.  In  the  case  of  factors  expressed  in  Arabic  numerals 
multiplications  like  the  following  may  be  carried  out  in  either 
of  two  ways. 

E.g.  32.34  =  9.81  =  729, 

or  32  .  3*  =  (3  •  3)(3  .3.3.3)  =  .32+*  =  3^  =  729. 

But  with  literal  factors  the  second  process  only  is  possible. 
E.g.  a^ '  a^  =  {a  '  a){a  •  a  '  a  ■  a)  =  a^+^  =  a^. 

WRITTEN  EXERCISES 

In  the  following  exercises  carry  out  each  indicated  multipli- 
cation in  two  ways  in  case  this  is  possible  : 


1. 

5  .  52. 

5. 

a2  •  a?. 

9. 

?2   .   ^  .  t\ 

2. 

52 .  51 

6. 

a^  •  X-. 

10. 

23  .  22  .  2\ 

3. 

32 .  32. 

7. 

x''  •  X*. 

11. 

3  .  32 .  33. 

4. 

7.73. 

8. 

f  -t^. 

12. 

22 .  23 .  22 .  2. 

94  ALGEBRAIC   EXPRESSIONS 

Illustrative  Problem.  To  multiply  2^  by  2%  k  and  n  being 
any  two  positive  integers. 

Solution.       2*  means  2  •  2  •  2  •  2,  etc.,  to  k  factors, 
and  2"  means  2  •  2  •  2  •  2,  etc. ,  to  n  factors. 

Hence,     2*=  •  2"  =  (2  •  2  •  2  •••  to  ^'  factors)  (2  •  2  ...  to  w  factors) 

=  2.2.2.2...toA:  +  n  factors  in  all. 
That  is,  2*  .  2"  =  2*+". 

The  preceding  examples  illustrate 

Principle  XV 

74.  Rule.  The  product  of  two  powers  of  the  same  base 
is  found  by  adding  the  exponents  of  the  factors  and 
mahing  this  sum  the  exponent  of  the  coinmon  base. 

But  exponents  are  added  in  multiplication  only  when  they 
apply  to  the  same  base.     Thus  2^  •  23=22+3=2^=32 ;  aP-  •  a^=a\ 

E.g.  The  product  of  2^  .  3"^  cannot  be  found  by  adding  the  exponents. 
It  must  be  done  as  follows  :  2^  .  32  =  8  .  9  =  72. 

ORAL  EXERCISES 

Perform  the  following  indicated  multiplications  by  means 
of  Principle  XV. 

1.  23  .  2\  7.  4«  .  4\  13.  x"-  .  3  0.-3. 

2.  o?  .  a\  8.  32-^  .  32^-.  14.  aj4  .  4  x\ 

3.  3^.35.  9.  52+".  52--.  15.  x^-bxK 

4.  a^  .  xK  10.  a"*  •  a".  16.  a^  •  2  a". 

5.  3*= .  3".  11.  C  '  c^-'.  17.  a/*  •2  a". 

6.  x^ .  cc".  12.  xf' '  xfi".  18.  a2"  •  3  a^"". 

Perform  the  following  multiplications  by  means  of  Prin- 
ciples II  and  XV. 

19.  x{x'^  -\-  x-\-l).  23.  y{S  7f-\-4:y*  —  f). 

20.  x^(x-\-l).  24.  x^{7  x*-5x^- 2x). 

21.  aXa^-a-^l).  25.  a'^it^  a- 2  a'^b  +  Aab^). 

22.  a3(a«-4a2  +  a  +  l).  26.  a2*(4a2*  -  a3^  +  a*^). 


PRODUCTS  OF  POWERS  OF  THE  SAME  BASE     95 

75.  Products  of  Monomials.  In  multiplying  together  mono- 
mials like  3  aH)G  and  2  ah'^cd  it  is  convenient  to  arrange  the 
factors  in  the  product  so  that  the  same  letters  are  associated 
together  and  likewise  the  numerical  coefficients.  This  we  are 
permitted  to  do  by  Principle  XIII. 

Thus,  3 a%c  x  2 abHd  =  (3  •  2) (a^  ■  a){b  -  h-) (c  ■c)d='6 a^b^c^d. 

Notice  that  in  the  product  the  exponent  of  each  letter  is  the 
sum  of  the  exponents  of  this  letter  in  the  factors,  and  the  numeri- 
cal coefficient  is  the  product  of  the  numerical  coefficients  of  the 
factors. 

E.g.  (2  aW)  (5  a^bH)  =  10  a*+VM^c  =  10  a^^c. 

This  is  a  convenient  rule  for  finding  the  product  of  two  or 
more  monomials. 

ORAL  EXERCISES 

Multiply : 

1.  3  ab  by  5  o'bK  16.  -  3  x'y^  by  2  xy\ 

2.  4.a^hy3xy\  17.  lOVy  by  -  1(P  ic^^. 

3.  2xyzhySx'yz.  18.  5^  .32  •  2^  by  5  •  3  •  2^ 

4.  6¥hj7  a¥.  19.  2  a^bc^  by  3  a¥c, 

5.  3  x^y"^  by  4  xy^.  20.  6  m^n  by  3  mn^. 

6.  5  a'^b^c  by  ab^c.  21.  2  ma^  by  3  a^. 

7.  2  6Va^  by  5  6*ca;.  22.  2  Z)^  by  7  6*. 

8.  3  a^b*c  by  aZxi''.  23.  4  x'^y-  by  3  x"/. 

9.  —  2  .t2?/2  by  5  a.y .  24.    3  my^  by  4  m^y^. 

10.  —  4  a^^c^  by  —  3  a¥c^.  25.  7  a.T^  by  4  aV. 

11.  —  3  t^u^  by  —  5  tu\  26.  5  a*y  by  ay. 

12.  a^x^by— 3aa;^  27.  4  ??i  V  by  mV. 

13.  5  a6c  by  —  abc.  28.  —  3  a^b*c  by  5  a^^^c^. 

14.  —  4  771^71  by  —  6  mji^.  29.  —  8  a^c^  by  —  4  aV;r 

15.  —  7  a^m?/  by  —  2  m^j/.  30.  -  4  xY  by  3  xV- 


96  ALGEBRAIC   EXPRESSIONS 

WRITTEN   EXERCISES 

Perform  the  following  indicated  multiplications : 

1.  (a -r  &)(«'  + 2>').  23.  (x""  +  x -tl)(x'' -  x -\-l). 

2.  (a  —  9)(a'^-{-b^).  24.  {x  +  y){x^— x^y -\- xy^— f). 

3.  (a  —  b)(a'^  —  b'^).  25.  {x  —  y){x^  +  xhj -^  xy^ -{- y^). 

4.  (a.'2  +  2/2)(a;2-/).  26.  (x''- xy -hy^){x''-\- xy -^  y^). 

5.  (x-y^')(x'^  —  y).  27.  (x"^ -{- 2  xy -j- y^)(x  -  yf. 

6.  (3  a^a^  -  /)(3  a^a;  +  2/^).  28.  (.r'  —  ?/2)(a;4  -f  a.^  ^  ,y4^)^ 

7.  (3a;2-2  2/2)(3a;2  +  2  7/2).  29.  {x' +  y'')(x^  -  xY-\- y'). 

8.  (2aa;2_3^^2^)(2aa;2+3  6?/2).  30.  (x^  +  2/2)(a;  + ?/)(a;-2/). 

9.  (7a;2  +  22/)(3x'2-22/).  31.  (.x'-2/)(a;2+a-2/+^2)(^^,^3)^ 

10.  (2a26-c2)(3a&2  +  c2).  32.  (a.'+2/)(aT2-a;?/+2/')G^''-2/'). 

11.  {3xy''-5a^y){x-y).  33.  (.r^ -2/3)(a^ +  ?/3). 

12.  (a+6)2(a-6).  34.  (x'  +  2  x-\-l){x''-2  x-^1). 

13.  (a-6)2(a+^).  35.  (.^;  +  2/)2(.^•  +  2/)2.     ■ 

14.  (a-{-by(a-by.  36.  (a;  +  2/)'(a;  +  2/)'- 

15.  {a'-\-a^b  +  ab''-\-b')(a-b).   37.  (a2  +  2  a6 +  62)(a -6)2. 

16.  (a  +  6-c)(a  +  &  +  c).  38.  (3  aa^  -  7)(3  aar^  +  7). 

17.  (3a;-2?/-l)(2a;4-2/).  39.  (H  ax' -y)(3  ax' -y). 

18.  (1  +  a  +  «')(!  -  a).  40.  (.^^  -  xy)(x'  -f-  a^V). 

19.  (l_a  +  a2_(^3)(l_^^^)_  41,  (9  a2  4- 3  aft  4- 2,2^(3  a  _  6). 

20.  (a  +  b) (a  —  &) (a^  +  b^).  42.    (.v  +  y  +  z) (a.-  —  y  —  z). 

21.  (a+ 6)(a2-a6  +  &2).  43.    (^a  +  b -c-\-dy. 

22.  (a  -  6)(a2  +  a6  +  6^).  44.    (x  —  y  —  z-vy. 

QUOTIENT   OF   TWO   POWERS   OF   THE   SAME   BASE 

76.   Illustrative  Problem.     To  divide  x^  by  x\ 

Since  by  §  50  tbe  quotient  times  the  divisor  equals  the  dividend,  we 
seek  an  expression  uhich  multiplied  by  x*  equals  x^. 

Since  by  Principle  XV  two  powers  of  the  same  base  are  multiplied  by 
adding  their  exponents,  the  expression  sought  nuist  be  that  power  of  a* 
whose  exponent  added  to  4  equals  6.  Hence  the  exponent  of  the 
quotient  is  6  —  4  =2.     That  is.  x^  -i-  x*  =  x^—*  =  x'^. 


QUOTIENT   OF   POWERS   OF   THE   SAME   BASE  97 

ORAL  EXERCISES 

Perform  the  following  indicated  divisions : 


1. 

2*  -=-  2^ 

8. 

5^3  ^  512. 

15. 

X'  -  0^2. 

2. 

23  -  22. 

9. 

724  ^  722_ 

16. 

^^4  H-  ^. 

3. 

24  _^  2. 

10. 

8^-8. 

17. 

m^  -j-  771. 

4. 

33  --  32. 

11. 

6'  -  62. 

18. 

?i«  ^  7^2^ 

5. 

3^ -^3. 

12. 

(i3  ^  a-. 

19. 

(20)4 -(20). 

6. 

34  --  32. 

13. 

a^  -=-  al 

20. 

(101)14^(101)13 

7. 

9"  ^  910, 

14. 

?>l^  -T-  ??i2. 

21. 

(41)^^(41)«. 

The  preceding  exercises  illustrate 

Principle  XVI 

77.  Rule.  The  quotient  of  two  powers  of  the  same  base 
is  a  power  of  that  base  whose  exponent  is  the  exponent  of 
the  dividend  minus  that  of  the  divisor. 

For  the  present  only  those  cases  are  considered  in  which  the  exponent 
of  the  dividend  is  greater  than  or  equal  to  that  of  the  divisor. 

Notice  that  Principle  XYI  does  not  apply  to  powers  of 
different  bases. 

E.g.  The  result  of  3^  -^  2-  does  not  equal  any  integral  base  to  the 
power,  4  —  2.  This  division  can  be  performed  only  by  first  multiplying 
out  both  dividend  and  divisor.     Thus,  3*  -f-  2^  =  81  -4-  4  =  20^. 

ORAL  EXERCISES 

Perform  the  following  indicated  divisions  by  means  of  Prin- 
ciple XVI : 

1.  2^-7-2'.  6.  x^"  ^  x^".  11.  x'^^^  ^  xf"-^^. 

2.  a'^o?.  7.  32<'-i  ^  3^-2.  12.  ?r^-M^. 

3.  34^-32.  8.  5"+^  -  o'»+2.  13.  (17)"--(17)i3 

4.  x^^x^.  9.  .x'"+* -i- .r°+2  14  5 a^^- 2.^2. 

5.  33'*-=-32^  10.  i-^^H-r.  15.  (12)^^(12)'. 


98  ALGEBRAIC   EXPRESSIONS 

WRITTEN  EXERCISES 

In  the  following,  use  Principles  III,  V,  and  XVI: 

1.  (3.  2^  +  5-23)- 22.  13.    {4.ax'--Sa''x)-^4.x. 

2.  (3.4^-5  .4^)--44.  14.    (8  a^ar*- 12  aar')H- 2  a?^. 

3.  {a'b-a'b^)^a\  15.    {12  a'x^  -  S  x^a-) -i- 4:  a\ 

4.  {4-.x^-^3x')^x\  16.    (16ali'3-8a;W)-=-2a;2. 

5.  •  (a^w^  -  bhn^)  h-  m\  17.    (24  icy  -|- 16  a^y'^)  -h  8  x"^. 

6.  (4a;2-5a;3  +  a;^)--.T2.  18.  (24  a^y  -  16  a^V)  h- 8  ?/2. 

7.  {3a'-\-9a'-2a')^a\  19.  (12  x^?/ -  16a:2^2)_j_4^^ 

8.  (12x^y  —  ll  xY + 5  .^'^)  -r- .t^.  20.  (a^"  +  x^)  -h  a;*. 

9.  (8a3  +  12a*)^4a'.  21.  (x"" -{- x^)  ^  x"". 

10.  (9a^  +  15a^)-=-3a;2.  22.    (x^""  —  x^"")  ^  x"". 

11.  (8a;^-16a;8)^8a:^  23.    (i/'^^  - /«) ^  ?/«. 

12.  (15  aj^  —  10  a.-^) -=- 5  a;2.  24.    (a-"*  —  a^"*) -v- a*". 

25.  (4a^-8a;^  +  16x«-20a^  +  12a;4)--4a.'^ 

26.  (4a;^— 8a;^"  +  16a.'^"  — 20a^^4-12aj*'»)H-4a;^". 

27.  (3  xy*  -  6  a;^^/^  +  12  x^y^  -  24  ?/^)  ^  3  ?/2. 

28.  (3  a;?/^"  -  6  a^y^'  +  12  xh/''  -  24  /»)  -=-  3  y'^ 

29.  (a'62_a«53_^^5^2_4^4^_^^4_ 

30.  (a'*62  -  a*^*63  +  a'^*&2  _  4  ^4a>)_^  ^^4a:_ 

31.  {x^Y  —  xY  -h  0^/  -  aJ^2/''  +  ^/)  -^  ^. 

32.  (a^^'^y  —  x^y^  +  a;^"^/^  —  ^'^V  +  -'^''^V)  "^  ^^• 

78.  Division  by  Monomials.  In  dealing  with  the  quotient  of 
two  monomials  the  indicated  division  may  be  written  in  the 
form  of  a  fraction  and  the  factors  common  to  dividend  and 
divisor  may  be  cancelled,  that  is,  divided  out  of  both  numerator 
and  denominator,  just  as  in  arithmetic. 

I    r  ft   Jt'O  w  firiO 

Example  1.    15  aWc  -i-  3  a^bx^y  = = '- — '■ — • 

3  d}bxhi        xhf 

Example  2.    12  a^a; -- 3  aa;  =  ^^^^' =  —  =  4  a. 

3aa;        1 


QUOTIENT  OF  TWO   POWERS   OF  THE   SAME   BASE      99 
ORAL  EXERCISES 

Give  the  following  quotients  in  their  simplest  forms : 
1     ^a^\  „      8  a'h'  ^3     48aVl\ 


3  mn  ciHf^c^  21  aj^^+ij/^+i- 

^      8.15.14  ,^     6-8.10  ,^     Wa'-b^' 

4.     •  10.     .  16. 

4.5.7  3.4.2  4  a"b'"' 

^     4:X^y^  12a?Y  _     14  ar'^Y' 

O.      — - — — •  XX. — •  IT.     '- — * 

2ry^  6  x'^y  7  x^y^ 

^     bo?f  ,^     10a^6V  ,^     21a^"62« 

0.     •  1a. *  10.     ► 

2  a;Y  5  o'bH  7  a-"^"' 


WRITTEN  EXERCISES 

Write  each  in  the  fractional  form  and  cancel. 
Divide  : 

1.  4.7.9a.'5by  2.3x'2.  5.  12 x^Y""  ^J  ^ ^V^^- 

2.  12.8.20a;8by2.4.5a;^  6.  Da^b^^chy  ab^&. 

3.  6  :x?y'^z  by  2  a;?/:^.  7.  10  x^h^^c^  by  2  a;6^c. 

4.  6^.54.a;3  by  62.52.0.-2.  g.  36 .t^?/^  by  g ^j^/S^ 

9.  4  aj22/3  —  3  a.*^?/2  by  xY- 

10.  18  a;V  -  12  x'f  +  6  xY  by  6  a;22/2. 

11.  49a^  +  21a3  — 7  a  by  7  a. 

12.  12  ax^y^  —  16  a^x^y^  +  8  a^a.'y  by  4  aicy. 

13.  2  a.-3«  +  4  ic^  -  8  .x'2»  by  2  .r". 

14.  6  a^2^-+i  +  12  a^^'^+i  _  10  0;"+^  by  2  a;"+i. 
15.4  .r"  -  6  «"6  -  10  a^c  by  2  .r^. 

16.   10  a^b''  -  20  a263  +  15  a^6^  by  5  a262. 


100  ALGEBRAIC   EXPRESSIONS 

79.  Negative  Exponents.  The  process  of  division  by  subtract* 
ing  exponents  leads  in  certain  cases  to  interesting  results. 

Thus,  X*  -r-x^  =  x^~^  =  scP,  which  seems  to  have  no  meaning, 
since  an  exponent  has  been  defined  only  when  it  is  a  positive 
integer.  The  exponent  zero  cannot  indicate,  as  in  the  case  of 
a  positive  integer,  how  many  times  the  base  is  used  as  a 
factor. 

We  know,  however,  that  x*  -i-  x*  =  1,  since  any  number 
divided  by  itself  equals  unity.  Thus,  any  number  with  the 
exponent  zero  is  equal  to  unity.  Hence  if  we  use  the  symbol 
0^,  it  must  be  interpreted  to  mean  1,  no  matter  what  riumher  is 
represented  by  x. 

Again  by  this  process,  x"^  -r-  x*  =  x^~*  =  x~^,  which  seems  to 
have  no  meaning,  since  negative  exponents  have  not  been  de- 
fined.    But  we  know  that  if  we  divide  numerator  and  denomi- 

X         1 

nator  by  x"^,  as  in  aiithmetic,  x^  -i-x'^  =  —  =  — 

X*      oc^ 

Hence  if  we  use  the  symbol  x~^,  it  must  be  interpreted  to 
mean  —  •         Similarly  x~^  =  —,  x~^  =  ~,  etc.     Negative  expo- 

nents  and  also  fractional  exponents  are  the  subjects  of  more 
advanced  work  and  are  considered  in  detail  in  the  Intermediate 
Course. 

ORAL  EXERCISES 

Express  the  following  quotients  by  mea.ns  of  negative  ex- 
ponents after  first  cancelling  any  common  factors,  as  in  §  78. 

T  1  ^         X  f.  o'> 

1.  — r.  5.      — .  y.      

x^  x^  d^W 

2.  1.  6.     «.  10.     ^. 

3.  i.  7.    ^.  11     '-^ 

4.  i.  8.     '-.  12.    HV. 
a^                                   c*  uY 


QUOTIENT   OF   TWO   POWERS   OF  THF  fi'A2/LE   liA^SE  :  iOtj 

ORAL  EXERCISES 

Express  the  following  in  the  fractional  form  : 

1.  a~\  5.    x-\  9.    xy-\ 

2.  b-\  6.    d-i".  10.    a-W. 

3.  a;-^  7.    a-^J-^.  11.    a-^^-^. 

4.  a-"*.  8.    x-^y-\  12.    x'^y-^. 

WRITTEN  EXERCISES 

By  means  of  negative  exponents  write  each  of  the  following 
without  using  the  fractional  form  : 


1. 


5  ax^  .      2  ah"^  «     4  h'^c 


3. 


a}h^  r        4  ^     Sxy 

•  5. -•  o. —* 

x^y*  a;'"?/"  xfy^ 


Express  the  following  without  using  negative  exponents : 

10.  ax-^y~\  13.    xy'z-^.  16.    x-^^'y-^h''^. 

11.  a-hjx-^.  14.    x-hjz-\  17.    a-^6  ^^-a 

12.  a-^y-^x\  15.    x'^y-^.  18.    a-'"6-2'»-3c. 

HISTORICAL   NOTE 

Exponents.  The  expression  "  power  "  is  used  by  Alkarismi  to  denote 
the  square  of  a  number.  Up  to  the  time  of  Vieta  it  was  customary  to  use 
different  letters  (if  letters  were  used  at  all)  to  express  the  square,  the 
cube,  etc.  of  a  number.  Thus,  if  B  represented  a  number,  Q  might  repre- 
sent the  square  of  it  and  C  the  cube  of  it.  Vieta  wrote  A^  A  quad,  ^4 
cube,  etc.,  for  A^  A^,  A^^  etc.  Harriot  wrote  aa  for  a-,  aaa  for  a^,  etc. 
Descartes  (1637)  established  the  usage  of  the  forms  a"-^,  a^,  etc. 

John  Wallis  (1616-1703)  explained  the  meaning  of  negative  and  of  frac- 
tional exponents.     Thus  he  wrote  .r-^  for  -,  x-'^  for  — ,  etc. 

X  x^ 

Sir  Isaac  Newton  (1642-1727)  used  exponents  of  any  magnitude.  Thus 
he  used  not  only  a-^,  x—'^^  x's,  but  also  such  expressions  as  x  .  These 
latter  need  not  be  discussed  here. 


102  ALGEBRAIC   EXPRESSIONS 

DIVISION  BY  A  POLYNOMIAL 
80.   Illustrative  Example.     Consider  the  product 

(x^  +  2xy-\-  y%x  -{- y)  =  x'ix  -\-y)-\-2  xy(x  +  2/)  +  fi^  +  2/). 

The  products  x'^(x  +  y),  2  xy{x  +  y),  and  y^(x  +  y)  are  called  partial 
products,  and  their  sum,  a;^  +  3  x~y  +  3  xy^  +  y^,  the  complete  product. 

Ill  dividing  x^  +  3  x-y  +  3  xy^  -\-  y^  by  x  -\-y,  the  quotient  must  be 
a  polynomial  such  that  when  its  terms  are  multiplied  hy  x  -\-  y  the 
results  are  these  partial  products. 

The  work  may  be  arranged  as  follows  : 
Dividend  or  product  =  x^  +  3  a;^^^  +  3  xy'^  +  y^  \x  +  y^  divisor . 

1st  product,  x-(x-\-y)  =  x^  +     x:^y |x-  +  2  xw  +  y\ 

Dividend  minus  1st  product  =  2  x'^y  +  3  xy'^  +  y^        [quotient. 

2d  product,  2xy(x  +  y)  =  2  x^y  +  2  xy'^ 

Dividend  minus  1st  and  2d  products  =  xy^  +  y^ 

3d  product,  y'^l^x  +  y)  =  xy'^  +  y^ 

Dividend  minus  1st,  2d,  and  3d  products  =  0 

Explanation.  Dividing  the  first  term,  x^,  of  the  dividend  by  the  first 
term,  x,  of  the  divisor  the  quotient  is  x"-.  Multiplying  this  term  of  the 
quotient  by  the  divisor,  we  obtain  the  first  partial  product,  x^  +  x^y. 

Subtracting  the  first  partial  product  from  the  whole  product,  x^  +  3  x"^y 
-f  3  xy'^  +  y^,  the  remainder  is  2  x-y  +  3  xy-  +  y^.  Dividing  the  first  term, 
Sx-^y,  of  this  remainder  by  x  the  quotient  is  2  xy.  The  product  of  2xy 
and  X  +  ?/  is  the  second  partial  product. 

In  like  manner  the  third  partial  product  is  xy'^  +  y^. 

After  subtracting  the  third  partial  product  the  remainder  i3  zero. 

Checking  Problems  in  Division.  Problems  in  division  may  be 
checked  by  substituting  any  convenient  values  for  the  letters. 
For  instance,  in  the  above  example,  if  x=\,  y  =  1,  we  have: 

Divisor  =x4-2/=l  +  l  =  2. 

Quotient  =  x'^  +  2  xi/  +  2/^  =  1  +  2  +  1  =  4. 

Dividend  =  x^  +  3  x^y  +  3  xy2  +  ?/3  =  i  4-  3  +  3  +  1  =  8. 

We  know  that  we  should  have 

Dividend  =  Divisor  x  Quotient. 
But  8  =  2x4.      Hence  the  correctness  of  the  division  is  shown. 

But  since  division  by  zero  is  impossible,  care  must  be  taken 
in  checking  not  to  select  sucli  values  for  the  letters  as  will  re- 
duce the  divisor  to  zero. 


Sir  Isaac  Newton  (1642-1727),  probably  the  greatest  mathe- 
matician of  all  time,  was  born  near  Grantham  in  Lincolnshire, 
England,  in  the  same  year  that  Galileo  died. 

He  entered  Cambridge  in  1661  and  began  at  once  a  career  of 
unequalled  productive  study.  The  binomial  theorem  was  one  of 
his  early  discoveries.  Later  Newton  laid  the  foundation  of  the 
calculus. 

His  chief  work,  the  Principia,  published  in  1687,  aimed  "to 
apply  mathematics  to  the  phenomena  of  nature."'  This  work 
placed  him  in  the  very  front  rank  for  all  time  among  mathe- 
maticians, physicists,  and  astronomers. 


DIVISION   BY  A   POLYNOMIAL  103 

Example.     Divide  x^— 7x^-\-2x*  —  l-{-Dxhj2x— l-{-x^. 

Solution.  We  first  arrange  both  dividend  and  divisor  according  to  the 
descending  powers  of  x.  [divisor. 

Dividend  or  product  =  2  x*  -{-    x^  —  1  x-  +  6x—  I       j;-+2 x—  1, 

Istproduct,  2  x2(x2+2 X-  1)  =2  x"*  +  4 x^  -  2  x2  2x-^-3x  +  l, 

Dividend  minus  1st  product  =  — 3x^  —  6x2-}-5x— 1  [quotient. 

2d  product,  -3  x(x2  +  2  x  -  1)  =-3x8-6x2  4-3x 
Dividend  minus  1st  and  2d  products  =  +  x^  -f  2  x  —  1 

3d  product,  1  •  (x^  +  2  x  —  1)  =  x^  +  2  x  —  1 

Dividend  minus  1st,  2d,  and  3d  products  =  0 

Check.     Substituting  x  =  2,  we  get  21  ^  7  =  3. 

81.  From  a  consideration  of  the  preceding  examples  the 
process  of  dividing  by  a  polynomial  is  described  as  follows : 

1.  Arrange  the  terms  of  dividend  and  divisor  according  to 
descending  (or  ascending)  powers  of  some  common  letter.  As  the 
division  proceeds,  arrange  each  remainder  in  the  same  ivay. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor.     This  quotient  is  the  first  term  of  the  quotient. 

3.  Multiply  the  first  term  of  the  quotient  by  the  divisor  and 
siibtract  the  product  from  the  dividend. 

4.  Divide  the  first  term  of  this  remainder  by  the  first  term  of 
the  divisor,  obtaining  the  second  term  of  the  quotient.  Multiply 
the  divisor  by  the  second  term  of  the  quotient  and  subtract,  obtain- 
ing a  second  remainder. 

5.  Continue  in  this  manner  until  the  last  remainder  is  zero,  or 
until  a  remainder  is  found  whose  first  term  does  not  contain  as  a 
factor  the  first  term  of  the  divisor. 

ORAL  EXERCISES 

Arrange  each  of  the  following  in  descending  powers  of  the 
letter  involved : 

1.  16a;-26.T2-t-3  +  llar'  +  a^. 

2.  8  a  -  5  a2  ^_  8  -  4  a^  _|.  2  a^  -  a'. 

3.  Sf  +  ^  +  ^y'-2y-y\ 

4.  422-18  +  62^  +  18  2. 


104  ALGEBRAIC   EXPRESSIONS 

WRITTEN  EXERCISES 

Divide  the  following.  Check  the  results  in  Examples  1  to  13, 
being  careful  to  substitute  such  numbers  for  the  letters  as  do 
not  make  the  divisor  zero : 

1.  a^-{-2ab  +  b^hj  a-\-b. 

2.  a'-2ab-}-b^hj  a-b. 

3.  a^-Sa'^b-j-3ab^-¥hy  a~b. 

4.  2  a^  +  2  x^y  —  4:X^  —  x  —  4  xy  —  y  hj  x  +  y, 
6.  a?  —  x^y  -f  xy^  —  y^  hj  x  —  y. 

6.  x^ -\- 4:  x^ -{- X  —  6  by  X -{- 3. 

7.  oi^  +  x-\-4:X^  —  6hyx—l. 

8.  x*  —  6:t^  +  2x'^-3x  +  6hyx-l. 

9.  x^  -\-3  xy^  +  3  x-y  -\- y^  hy  x"^ -\- y^ -\- 2  xy. 

10.  x^  _  8  a^  _^  75  by  a;  -  5. 

11.  2  a^  _^  19  a^b  +  9  ab"^  by  2  a  +  b. 

12.  a^  -{-  y^  by  X  +  y. 

13.  xi^  —  y^  by  x  —  y. 

Divide  the  following : 

14.  X*  +  a^y  +  xy^  -^y*  by  x-\-y. 

15.  X* -{-y*  -\-  a^y^  by  x"^—  xy  -\- y^. 

16.  x^  —  y*  by  X  —  y. 

17.  a^  +  a^-12aj2^14a;-4  by  a;2-3a;  +  2. 

18.  2a.'^  +  lla.-3-26a;2  +  i6a;-3  by  ic^  _^  7  a;  -  3. 

19.  x'" -{- 5  x*y  +  10  x^y^  +  10  x'^y^  -{- 5  xy* -\-  y^  by  a;^  -f-  2  xy  +  ?/2 

20.  x''  +  10  x"  -  x'  -21  x"  -30  X  -200  by  a;2_4a;_io. 

21.  3  a;2  _  4  a;?/  4-  8  0^2  —  4  7/2  +  8  2/;^  —  3  22  by  a;  -  2  ?/  +  3  2. 

22.  9  r^s^  -  4  rH^  +  4  ?-s^2  _  ^2^2  ^y  3  rs  -  2  rt  +  M. 

23.  9  a2?>2  4.  iG  3^2  _  4  ^2  _  3(5  ^2^,2  ^y  3  a6  +  <>  6.f  -  2  a  -  4  a;. 

24.  a^  -h  ^"^y  +  ^'^2:  —  aj//2  —  yh  —  yz^  by  x"^  —  7/2;. 

25.  a^  -  a^b"^  +  a^6  -|-  a?b  -f-  ai^  -  6^  by  a}  +  t(6  -  6^. 


DIVISION   BY   A   POLYNOMIAL  105 

82.  Division  not  Exact.  In  case  the  division  is  not  exact,  the 
remainder  may  be  placed  over  the  divisor  in  the  fractional  form 
as  in  arithmetic. 

Example.     Divide  20  a^  _  4  +  18  a^  -f- 18  a  -  19  a' 

by  2a2-3a  +  4. 
Solution.     Arranging  dividend  and  divisor  according  to  the  descend- 
ing powers  of  a,  we  have  [divisor. 
Dividend  or  product :     18  a*  —  19  a^  4.  20  a'-^  +  18  a  —  4    2  g'^  —  3  q  +  4, 

1st  product:  18  a^  -  27  g^  +  36  a^ 9  a2  +  4  a  -  2, 

Dividend  minus  1st  product :      8  a^  —  16  a"^  +  18  a  —  4  fquotient. 

2d  product :  8  g^  -  12  a^  4-  16  a 

Dividend  minus  1st  and  2d  products  :    — 4a2+    2a  —  4 
3d  product :  — 4a2_|.    6a  —  8 

Dividend  minus  all  products :  —  4  a  +  4 

Since  2  a^  is  not  contained  in  —  4  a,  the  division  ends  and  —  4  a  +  4  is 
the  remainder.  As  in  arithmetic  V7e  write  this  as  the  numerator  of  a 
fraction  whose  denominator  is  the  divisor.  Hence,  the  complete  result  is 
4-4a 


9a2  +  4a-2  + 


2  a2  -  3  a  +  4 


WRITTEN  EXERCISES 

In  the  following  express  the  remainders  in  the  fractional 
form: 
Divide : 

1.  4  ic^  -  2  .^2  4-  6  a^  +  4  a.'  -  7  by  2  x"-  -  x  +  2. 

2.  8  a^  -  4  +  7  a^  +  2  a^  -  2  a2  by  a^  -  3  a  +  1. 

3.  3-4.X  +  1  x'-  -3x^-\-x'  by  x  -  4. 

4.  2'X^-6  +  2x^-6x  +  6x'  by  x""  +  o. 

5.  lSx-7  x''  +  Sx'-^3x^-\-Uhy  x''-\-S. 

6.  21  -  7  x'  -  9  ic^  +  8  a^2  _|_  ^^4  \^y.  ^.3  _^  j 

7.  12a^-4a;2  +  8a;  +  16aj^by  .r-  +  .i-  +  4. 

8.  35a-14+21a2  +  7a3by  a-o. 

9.  16  x''  -  14  x'  +  2  x^  -  X  +  2  by  2  x"-  4  x^  +  8. 

10.  2  0^4  _  11  .^.3  _^  26  .^2  +  18  ic  +  3  by  x""  -7  x-{-  3. 

11.  6x^-2x^-\-2  x'  -  ^2  +  x  -  6  by  3  a;2  -  4  a;  +  2. 


106  ALGEBRAIC   EXPRESSIONS 

83.  Division  by  Detached  Coefficients.  When  both  dividend 
and  divisor  are  arranged  in  descending  powers  of  the  same 
letter,  the  work  of  dividing  by  a  polynomial  may  be  shortened 
by  omitting  the  letters  and  writing  only  the  coefficients. 

Example.     Divide  2  ocf^  -}-  j^  —7  ay^  -{-  5  x  —  1  by  x^  -{-2x  —  l. 

1  +2-1 


Writing  coefficients  only,    2  +  1  —  7  +  5 

2  +  4-2 


2-3  +  1 


-3-5+5- 
-3-6+3 

■  1 

+  1+2- 
+  1+2- 

1 
1 

Since  dividend  and  divisor  are  arranged  in  descending  powers  of  x,  we 
know  that  the  quotient  will  be  so  arranged  also.  Since  the  first  term  of 
the  quotient  contains  x^  -^  x^,  the  quotient  starts  with  x^  and  is  then 

2a;2-3x  +  l. 

This  example  is  worked  without  detached  coefficients  on  page  103. 

If  any  terms  are  lacking  in  the  dividend  or  divisor,  zero 
must  be  written  as  the  coefficient  of  each  such  term,  as  in  the 
following  example. 

Example.     Divide  ic^  +  cc^  +  1  \)jx'^—x-\-  1. 

Since  the  third  and  first  powers  are  lacking  in  the  dividend,  we  write 
zero  in  place  of  each. 


1+0+1 +0+1 
1-1+1 


1-1  +  1 


1  +  1  +  1 


+1+0+0+1 

+1-1 +1 
+1-1+1 
+1-1+1 

Hence,  the  quotient  starts  with  x"*  -;-  x'^  =  x^,  and  is,  then,  x^  +  x  +  1. 
WRITTEN  EXERCISES 

Divide  by  detached  coefficients  Examples  3,  6,  7,  8,  10,  12, 
13,  16,  and  20,  page  104. 

Note. — Division  by  detached  coefficients  is  much  used  in  certain 
topics  in  higher  algebra  where  the  work  is  still  further  abbreviated  by  a 
process  called  synthetic  division. 


REVIEW   QUESTIONS  107 

REVIEW  QUESTIONS 

1.  Make  a  diagram  to  show  how  to  multiply  (7+4)  by  (11  +  8) 
without  first  uniting  the  terms  of  the  binomials.  Multiply 
(a +6)  by  (c+d)  in  the  same  manner. 

Multiply  (12—3)  by  (9  —  7)  in  two  ways  and  compare  results. 
State  the  principle  by  which  two  polynomials  are  multiplied. 

2.  Describe  a  convenient  manner  of  arranging  the  work  in 
multiplying  polynomials.  What  kind  of  terms  in  the  product 
are  placed  in  the  sam  e  column  ?  Find  the  product  of7a;— S^Z+l 
and  2  x—S  y  —  S,  arranging  your  work  this  way. 

3.  State  Principle  XIII.  How  do  you  arrange  the  factors 
in  multiplying  3  a^b,  2  ab'^,  and  5  a*b^  ? 

4.  Define  a  positive  integral  exponent.  Explain  the  dif- 
ference between  an  exponent  and  a  coefficient. 

5.  Under  what  circumstances  are  exponents  added  in  mul- 
tiplication ?  State  Principle  XV.  Use  this  principle  to  show 
that  (ay  =  a\  (a'y=:a'-. 

6.  Under  what  circumstances  are  exponents  subtracted  in 
division  ?     Sta+e  Principle  XVI. 

7.  What  is  meant  by  x~^?  How  may  -  be  written  with 
a  negative  exponent  ? 

8.  How  may  division  by  a  polynomial  be  shortened  by  use  of 
detached  coefficients  ? 

Add  Principles  XIII,  XIV,  XV  to  your  list  expressed  in 
symbols  •. 

{a-b'C  =  acb  =b'Ca,  etc. 

xni     .         ,  ' 

[a-b'C  =  aoc  =  a-oc 

XIV  flw.a"  =  flw+w 

XV  flw  -7-  a«  =  a'"—" 


CHAPTER   VI 


SPECIAL  PRODUCTS   AND  QUOTIENTS 

There  are  certain  products  and  quotients  the  formulas  for 
which  should  be  memorized.  Some  of  these  are  collected  in 
this  chapter. 

THE   SQUARE   OF   A   BINOMIAL 

84.  The  square  of  the  sum  of  two  numbers  is  found  by  ordi- 
nary multiplication  as  in  §  67. 

E.g.  {a  +  6)  (a  +  6)  =  cfi  +  ah  +  ah +  h^  =  a^ +  2  ah -V  62. 


Hence, 


6a 

h^ 

o« 

ah 

(a  +  6)2  =  a2  +  2  a6  +  b\ 

This  product  is  illustrated  in  the  accom- 
panying figure. 

Translated  into  words,  this  identity  is : 

The  square  of  the  sum  of  two  numbers  is 
equal  to  the  square  of  the  first,  plus  twice  the 
2Jrodiict  of  the  two  numbers,  2^lus  the  square 
of  the  second. 


ORAL  EXERCISES 

Read  at  sight  the  squares  indicated  by  the  following : 


1.  (x-^yy. 

2.  (m  +  ny. 

3.  (x-{-iy. 

4.  (a;  4-2)2. 

5.  (2/ +  3)2. 


6.  (a +4)2. 

7.  (2o  +  l)2. 

8.  (2  a; +  1)2. 

9.  (3  +  a)2. 
10.  (3  a +  1)2. 


11.  (4  +  a;)2. 

12.  (l  +  3a;)2. 

13.  (2  +  3a)2. 

14.  (2  a +  3)2. 

15.  (3  a +  4)2. 


By  the  above  formula  we  may  square  any  binomial  sum. 
E.g.     (3x  +  2yy=  (3a;)2+2.  (Sx)(2y)  +  (2yy2  =9  x^+ 12  xy-\- 4  y'^ 


108 


THE   SQUARE   OF   A  BINOMIAL  109 

WRITTEN  EXERCISES 

Write  the  squares  indicated  by  the  following : 

1.  (2  a +  3  6)2.  6.    (4c4-5a)2.  11.  (12  +  7  a)^. 

2.  (3  2/ +  8)2.  7.    (8a  +  2  6)2.  12.  (10  +  3^)2. 

3.  (5  a;  4- 4)2.  8.    (4??i  +  3n)2.  13.  (16a  +  by. 

4.  (3a  +  7c)2.  9.    (6x-{-5yy.  14.  (Sx-\-8  7jy. 

5.  (2  a +  9  6)2.  10.    (a; +  12)2.  ^5^  (12  a  +  7  6)2. 

85.  Similarly,  we  obtain  the  square  of  the  difference  of  two 
numbers :  (a  ^  bf  =  a'- -2  ab  +  bK 

That  is,  the  square  of  the  difference  of  two  numbers  is  equal  to 
the  square  of  the  first,  minus  twice  the  product  of  the  two  numhersy 
plus  the  square  of  the  second. 

Example.  By  means  of  this  formula,  find  the  square  of 
a  —  3  6. 

Solution,     (a  -  3  6)2  =  a^  ~  2  •  a(3  6)  +  (3  6)2  =  a'^-6ab  +  9  b\ 

ORAL  EXERCISES 

Read  at  sight  the  squares  indicated  by  the  following: 

1.  (x  -  y)\  5.    (x  -  3)2.  9.    (2  -  x)\ 

2.  {m-nf.  6.    (a  -  4)2.  10.    (3  -  a;)2. 

3.  {x  -  1)2.  7.    (a  -  5)2.  11.    (4  -  xy. 

4.  {x  -  2)2.  8.    (2  a  -  1)2.  12.    (1-2  ay. 

WRITTEN  EXERCISES 

Write  the  squares  indicated  by  the  following : 

1.  (4a;-3?/)2.  6.   (4c-oa)2.  11.  (12-7a)2. 

2.  {by-'dxy.  7.    (8  a -3)2.  12.  {Ix-^yy. 

3.  (2a-3&)2.  8.    (7 .T- 4^)2.  13.  (8a -36)2. 

4.  ipx-  3)2.  9.    (3  X  -  7  yy.  14.  (12  a  -  13)2. 

5.  (4-3  a;)2.  10.    (a6  -  4  cy.  15.  (9  a  -  7  6)2. 


110  SPECIAL   PRODUCTS   AND   QUOTIENTS 

86.    Squaring   Polynomials.     By   means    of   the  formulas   in 
§§84  and  85,  we  may  square  any  polynomial. 

1.  (a  +  6  +  cy'  =  [(a  +  5)  +  cj'  =  (a  +  by'  +  2(a  +  b)c  +  c^ 

=  a^  +  2  ab  +  b'^  -\-2  ac  -\-  2  be  +  c-. 

2.  ia+r-s  +  ty'  =  l{a  +  r)-is-t)y 

=  (a  +  ry  -  2(a  +  r) (s-t)  +  (s-  ty- 

=  a-+2  ar+r"2-2  (as-at  +  rs—rt)  +  s"^— 2  st  +  t- 

=  a2+2  ar+r--2  as-i-2  at-2  rs-\-2  rt  +  s'^—2  st+t^. 


WRITTEN  EXERCISES 

1.  (a  +  5  -  c)2.  7.  [(a  -  3)  -  2(6  +  c)]^, 

2.  (a-6  +  c)2.  8.  [(m  4- 3)  -  Oi  +  a)]2. 

3.  (a  —  6  —  c)2.  9.  (a  —  b-\-c—  df. 

4.  (a  -  6  +  3)2.  10.  (x  -y  +  z-  3)1 

5.  {^a-2h  -\-ryf.  11.  (a-.i-  +  Z>-c)2. 

6.  [7a;-(4r-s)]2.  12.  {2-x  +  y  +  z)\ 


PRODUCT   OF   THE   SUM   AND   DIFFERENCE  OF   TWO   NUMBERS 

87.    Examples.     Find  the  products  : 

{x  +  5) (a;  —  5)  and  (.t  -\-  a){x  —  a). 

Solutions 

ic  +  6  X  +  a 

x  —  b  X—  a 

x^  +  5  X  x:'  4-  <^/.>" 

—  5a;— 25  —  ax  —  (r^ 

^  I^  x'-^  -  a2 

In  each  of  these  examples  one  factor  is  the  sum  of  two 
numbers  and  the  other  factor  is  the  difference  of  the  same 
numbers.  In  each  case  the  product  is  the  difference  of  the 
squares  of  the  numbers.     This  is  expressed  by  the  formula 

{x  H-  a){x  —  a)  =  jr^  —  a.-. 

That  is,  the  product  of  the  sum,  and  dijferoice  of  tico  numbers 
is  equal  to  the  difference  of  their  squares. 


PRODUCT   OF   THE    SUM   AND   DIFFERENCE 


111 


ORAL  EXERCISES 


Read  the  following  products : 

1.  (a  +  l)(a-  1). 

2.  (a4-3)(a-3). 

3.  {k-  h){k  +  h). 

4.  (3  -  a;)(3  +  x). 

5.  (2a  +  36)(2a-36). 

6.  (a  +  2  6)(a  -  2  6). 

7.  (2  6-l)(2  5  +  l). 

8.  (l  +  3x)(l-3x). 


9.  (l_7  2/)(l+7  2/). 

10.  (ct  — 4  6)(a  + 4  6). 

11.  (6a-36)(6a  +  35). 

12.  (7-9a)(7  +  9a). 

13.  (2c  +  l)(2c-l). 

14.  (3  a  +  &)(3a  —  6). 

15.  (5^'  +  3/i)(oA;-3/i). 

16.  (9  77l  +  3  7l)(9?7l  —  3  7l). 


By  means  of  the  above  formula  a  product  may  be  written  at 
once  whenever  the  factors  can  be  expressed  as  the  sum  and 
difference  of  the  same  two  number  expressions. 

E.g.  {x  +  y  -  z)(x  +  y  +  z)  =  [{x  +  y)  -z-llix  -^  y)+  z] 

=  a-/-  +  2  xy  +  y-  —  z^. 


WRITTEN   EXERCISES 

In  this  manner  form  the  following  products.      Verify   the 
first  five  by  formal  multiplication. 

1.  (4a  +  5  6)(4a-5  6). 

2.  (5  -  6  ^2)  (5  _p  6  ^2)^ 

3.  (3.x-22/)(3x  +  27/). 

4.  (^  —  y^)  (oc^  ■{-  y^y 

6.  (16a26'-3c)(16a263+3c). 

7.  (24X+12?/)  (24^-122/). 

8.  [x-^(y-z)]lx-(y-z)y 

9.  (x""  +  I/")  (cC"  —  y"). 

10.  [c-(a-5)][c  +  (a-6)]. 

11.  [a;-(2/+2;)][x  +  (?/+2;)]. 


12.    ( 

[a  -{-  b  -\-  c)  {a  —  b  —  c). 

13.    ( 

'a  +  b  —  c){a  —  b  -{-  c). 

14.    ( 

[a  —  b  -\-  c)  {a  —  b  —  c). 

15.    ( 

[r  —  y  —  z)  (r  —  y  -\-  z). 

16.    ( 

[a  +  b  +  c)  {a  -\-  b  —  c). 

17.    ( 

[x  +  2y  +  z){x-^2y-z). 

18.    ( 

'x-2y  +  z){x-{-2y-zy 

19.    ( 

[x  —  2y  —  z)(x  4-  2  y  -f-  z). 

20.    ( 

;a_f.26-c)(a-25-f  c). 

21.    ( 

[a-2b-c){a-2b  +  c). 

22.    ( 

2a+36-4c)(2a-36+4c). 

112  SPECIAL   PRODUCTS  AND   QUOTIENTS 

BINOMIALS   WITH  FIRST  TERMS   ALIKE 

88.    Examples.     Find  the  products  : 

(xi-2)(x  +  3)  (x-\-A)(x-7) 

{x  +  ^)(x  -  2)  (x-  5)(x  -  3) 

Solutions 


x  +  2 

a;  +  4 

x  +  S 

x-7 

oc^  +  2x 

X+  4:X 

3x  +  6 

-  7  X  -  28 

a;2  +  5x+6 

x2-3a;-28 

In  like  raanne^,      (x  +  5)  (x  —  2)  =  x^  +  3x  —  10, 
and  (x  -  5)  (x  -  3)  =  x^  -  8 x  -h  15. 

From  a  study  of  these  examples,  we  deduce  the  formula 

(jr  +  a)(A'  +  6)  =  jr^  +  (a  H-  b)x  +  ab. 

That  is,  the  product  of  two  binomials  having  the  first  terms 
alike  is  equal  to  the  square  of  the  first  term,  phis  the  first  term 
multiplied  by  the  algebraic  sum  of  the  last  terms,  plus  the  pi'oduct 
of  the  last  terms. 

ORAL  EXERCISES 

Eead  the  products  of  the  following : 

12.  (x-l){x-2), 

13.  (x-2)(x-3). 
{x-^2)(x-S). 
(x-{-S)(x-\-^). 
(x-3){x-^4.). 
{x  +  S){x-4.). 
(^x-3)(x-4). 
(x  +  4){x -\- 5). 
(.^•-4)(a;  +  ^)). 
(a;  +  4)(.^•-^)). 

22.    (;c-4)(a;  — 5). 


I. 

{x  +  l){x  +  2). 

12. 

2. 

(x-^l){x+3). 

13. 

3. 

(a;  +  l)(a;+4). 

14. 

4. 

{x  +  l){x  +  5). 

15. 

5. 

(x-l)(x  +  2). 

16. 

6. 

(x~l){x-{-3). 

17. 

7. 

(x-l){x  +  ^). 

18. 

8. 

(x-l){x-i-5). 

19. 

9. 

{x  +  2){x  +  3). 

20. 

10. 

{x-2){x  +  3). 

21. 

11. 

{x-{-2){x-4). 

22. 

BINOMIALS   WITH   FIRST   TERMS   ALIKE 


113 


WRITTEN  EXERCISES 


Find  the  following  products  : 


1. 

{x-\-7){x-{-3). 

12. 

(2a-l)(2a-3). 

2. 

(u;  +  9)(a^  +  6). 

13. 

(4a;  +  3)(4cc-h5). 

3. 

(2/  +  6)(2/-2). 

14. 

(4a;-3)(4a;  +  5). 

4. 

(y-S){y  +  3). 

15. 

(4a;4.3)(4a;-5). 

6. 

(c-4)(c-2). 

16. 

(4a;-3)(4x-5). 

6. 

(a-8)(a  +  10). 

17. 

(5y+6)(o?/  +  4). 

7. 

(a  +  7)(a  +  6). 

18 

(52/  +  6)(52/-4). 

8. 

(a-7)(a  +  6). 

19. 

(5  7/-6)(5  2/  +  4). 

9. 

(ab+3)(ab-\-7). 

20. 

(5y-6)(5  2/-4). 

10. 

(ab-5){ab-3). 

21. 

(ab  -f  6)(a6  +  7). 

11. 

(2a  +  l)(2a  +  3). 

22. 

{ab-3)(ab-\-7). 

In  the  formula    (a^  +  a){x  +  6)  =  a;^  +(a  +  b)x  -\-  ab,  replace 
a  and  b  by  the  following  values  and  simplify  the  results : 

23.  a  =  5,b  =  S.  25.    «  =  6,  6  =  -  11. 

24.  a  =  S,b=  —  T.  26.    a  =  -  5,  6  =  —  7. 

27.  Find  the  square  of  42  by  writing  it  as  a  binomial,  40  +  2. 

28.  Square  the  following  numbers    by  writing   each    as    a 
binomial  sum :  51,  53,  93,  91,  102,  202,  301. 

29.  Find  the  square  of  29  by  writing  it  as  a  binomial,  30  —  1. 

30.  Square  the  following  numbers  by  first  writing  each  as  a 
binomial  difference :  28,  38,  89,  77,  99,  198,  499,  998,  999. 

31.  Find  the  product  of  41  and  39,  first  indicating  the  prod- 
uct thus,  (40  +  l)(40-l). 

32.  Find  the  following   products    by  writing  each  pair  of 
factors  as  the  sum  and  difference  of  two  numbers : 


(1)  62  .  58. 

(2)  27  .  33. 


(3)  53  .  47. 

(4)  102.98. 


(5)  17  .  13. 

(6)  99  •  101. 


114  SPECIAL   PRODUCTS  AND   QUOTIENTS 

THE   SQUARE  OF   A  TRINOMIAL 

89.  Example.  By  multiplication  find  the  square  of  a-f-6  +  c, 
and  reduce  the  result  to  simplest  form. 

jSolution.  By  means  of  the  formula  in  §  84  we  can  perform  this  multi- 
plication by  a  short  method ;  namely, 

(a  +  &  +  c)2=[(a  +  6)+c]'^ 

=  a^  -{-  2  ab  +  b'^  +  2  ac  +  2  be  -{■  c-. 

How  many  terms  are  there  in  the  product  ?     How  many  are 
squares  ?     How  many  are  of  the  type  2  ah  ? 
From  this  we  get  the  following  rule : 

Tlie  square  of  a  trinomial  consists  of  the  sum  of  the  squares  of 
its  terms  plus  twice  the  product  of  each  term  by  each  succeeding 
term. 

In  symbols  this  is 

(a  +  6  +  c)2  =  a2  +  A^  +  c2  +  2  a6  +  2  ac  +  2  6c. 

The  above  rule  may  be  used  to  find  the  square  of  a  —  h-\-c 
as  follows : 

a-hJ[.c  =  a  +  (-  6)  +  c  =  a2  +  (_5)2  +  c2+2a(-6)+2  ac  +  2  {-b)c 

=  a2  +  62  +  c2  -  2  a6  4-  2  ac  -  2  be. 
Similarly,  (2  «  +  &  -  3  c)"' 

=  (2  a)2  +  62  +  ( _  3  c)2  +  2(2  a)&  +  2(2  a)  (-  3  c)  +  2  6( -  3  c) 
=  4  a2  +  52  _^  9  c2  +  4  a^  _  12  ac  -  6  be. 

Hence,  In  the  square  of  a  trinomial,  the  squared  terms  are  all 
positive  and  the  double  products  are  negative  luhen  one  factor  is 
negative,  otherwise  they  are  positive. 

ORAL  EXERCISES 

Give  the  squares  of  the  following : 

1.  x-\-y-\-z.  5.  a-\-b-{-2.  9.  a  — 25  +  3. 

2.  x-^y  —  z.  6.  a-\-b  —  2.  10.  a-\-2b  —  S. 

3.  x  —  y~z.  1.  a  —  b  —  2.  11.  a  —  2b  —  '?>. 

4.  x  —  y-\-z.  8.  a- 6  +  2.  12.  a +  2  6  +  3. 


THE   SQUARE   OF  A  TRINOMIAL  115 

ORAL  EXERCISES 

Give  the  squares  of  the  following : 

1.  3  —  y  —  z.  7.  —x-{-y-\-z.  13.  x-\-y-{-l. 

2.  3  —  x-j-y.  8.  —  X  —  y -\- z.  14.  x-\-y  —  l. 

3.  3  +  «  +  6.  9.  —x  —  y  —  z.  15.  x  —  y-l. 

4.  a  —  3  +  6.  10.  —  a  +  6  +  2.  16.  x  —  y  +  1. 

5.  a -I- 3  — 6.  11.  —a  —  b  +  2.  17.  —  x  —  y -\- 1. 

6.  a  —  S—b.  12.  —a-\-b  —  2.  18.  —x-\-y-l. 

WRITTEN  EXERCISES 

Find  the  squares  of  the  following : 

1.  Sa  —  2b-\-c.  13.  2  a'^b -\- ab^ -^  ab. 

2.  3a-\-2b-c.  14.  a6  +  aW  +  a^bK    . 

3.  4  a  — 3  6  + 2  c.  15.  2a;2  +  a:-f-l. 

4.  4a_3  6-2c.  16.  3a;3_^2a;2-|-a;. 

5.  2a +  4  6  — 6  c.  17.  aic  +  5?/ -f  cs;. 

6.  x  —  4:y-\-Sz.  IS.  aa^  +  6a;2  +  ca;. 

7.  3  a^2_|.2a;2/  +  3?/2.  19.  a-b'' ~  b^(^  -  a''c\ 

8.  a2  — 52_^c2.  20.  a V  -  ar^?/2  +  2/222, 

9.  2a2  +  352  +  c2.  21.  a2x' +  ay  +  ^''?/. 

10.  ab  +  ac  +  6c.  22.    3  a  —  4  a;  +  3  b. 

11.  2a2-362  +  4c2.  23.    5a -66  + 7c. 

12.  a262 -j- ?)V  +  c^al  24.   3a;-5y-9z. 

25.  Find  the  square  of  a  -\-  b  -^  c -\- d  by  writing  it  in  the 
form  r(a  +  6  +  c)  4-rf]-  Study  this  product  and  make  a  rule 
for  squaring  a  polynomial  of  four  terms. 

Find  the  squares  of  the  following : 

26.  x-\-y-\-z-{-  ic.  29.  m  +  2  ?i  +  3  r  +  s. 

27.  a- b-\-c-d.  30.  2  a  -  3  6  +  c  -  cZ. 

28.  a  — 2  6  — c  — 2d.  31.  3x-{-2  y  —  z -{-w. 


116 


SPECIAL  PRODUCTS   AND   QUOTIENTS 


CUBE  OF   A   BINOMIAL 

90.  Example.  Find  the  cube  of  a-\-b  by  first  finding  the 
square  of  a  -\-  b  and  then  multiplying  this  result  by  a  4-  b. 
How  many  terms  are  there  in  the  product? 

From  this  we  get  the  following  formula : 

(a  -\-by  =  a'-\-3  a'b  +  3  06^  +  b\ 

In  words  this  is : 

TJie  cube  of  a  binomial  is  equal  to  the  cube  of  the  first  tei^m, 
plus  three  times  the  square  of  the  first  term  multiplied  by  the  sec-' 
and,  plus  three  times  the  first  term  multiplied  by  the  square  of  the 
second,  plus  the  cube  of  the  second  term. 

The  above  rule  may  be  applied  to  find  the  cube  of  a  —  b,  thus 

(a-  6)3=  [rt+(-  6)]3  =  a3  +  3a2(_5)  +  3a(-6)2+  {-by 

=  a3  _  3  a^b  +  3  ab-^  -  b'K 

Which  terms  in  the  product  are  negative  and  why? 
Translate  into  words  the  formula : 

(a  -by  =  a'-3  a'b  -^Zab^-  b\ 


EXERCISES 

Find  the  following  cubes.     Read  the  first  six  at  sight. 


1. 

2. 

7. 

8. 

9. 
10. 
11. 
12. 
13. 


x  +  yf.  3.    {c  +  df.                    5.    (?7i+?i)3. 

X  —  2/)^  4.    (c  —  dy.                   6.    (m  —  ny. 

2  aj  -h  3  yy=(2  xy  +  3(2  a^)^  .  3  ?/  +  3  .  2  a:(3  yy+  (3  yy. 
2x-3yy.  14.  {x-iy. 
x-\-2yy.  15.  (a; +1)3. 
x-?>yy.  16.  {2x  +  ay. 
'?>a-by.  17.  (2x-ay. 

3a +  26)3.  18.  (6  +  6)3. 

3  a -2  6)3.  19.'  (2  a -5  6)3. 


Historical  Note.  The  Hindus  and  Arabs  had  the  formulas  for  (a  +  6)^ 
and  (a  +6)^-  Of  course  their  formulas  were  not  stated  in  terms  of  our 
symbols.     Vieta  used  the  formula  for  {a  -f-  by. 


QUOTIENTS   DERIVED   FROM   SPECIAL  PRODUCTS     117 
QUOTIENTS  DERIVED   FROM   SPECIAL   PRODUCTS 

91.    From  the  special  products  given  in  §§  84-88,  we  may  at 
once  derive  certain  special  quotients  by  observing  that  if  either 
factor  be  used  as  a  divisor  of  the  product,  the  other  factor  is  the 
quotient. 
Thus,  from 

(a  +  6)(a  +  &)  =  a'  +  2  a6  +  h\  (1) 

{a-h){a-h)  =  a}-2  ah  +  h\  (2) 

(a-\-h){a-h)  =  o}-h\  (3) 

{x  +  o)  {x  +  h)  =  x"^  +  (a  +  h)x  -{-  ah,  (4) 

we  derive    ^^^2^206  +  6^)  -  (a  +  6)  =  a -f  6,  (1) 

{a}  -2ab  -h  b')  ^  {a  -  b)  =  a  -  b,  (2) 

{a''-b')-^(a-b)  =  a-j-b 


(^a^^b')-^{a-\-b)  =  a-bi'  ^^ 

[jr2  +  (a  4-  6)jf  -h  a  6]  -  (jr  +  a)  =  jr  +  6  1 

Ix'  +  (a  -{-  b)x  +  ab]  ~  (x  +  b)  =  X  -{-  a]  ^  ^ 


EXERCISES 

Give  the  following  quotients  orally  if  possible : 
Divide:  1.   x^-\-2xy -{-y^  hy  x +  y. 

2.  x"^  -{- A  xy  -^  4:  y~  by  x  -i-2y. 

3.  x"^  —  6  xy  -\-9  2/2  by  x  —  3  y. 

4.    x^  —  y^  by  x  —  y.  7.    1  —  9  c^  by  1  —  3  c. 

6.   x"^  -  y*  hy  x"^ -{- y\  8.    Oc^— lby3c  +  l. 

6.    25a2-16  62by5a-4  6.         9.    {a -\-hy-c'^  hy  a +  h - 

10.  (a  +  hy  -  c2  by  a  +  ?>  +  c. 

11.  x^  —  {y-]-zyhyx  —  y  —  z. 

12.  a.'2-(2/  +  2;)2by  .V  +  2/  +  2;. 

13.  x:^  +  5x-{-6  by  .T  +  3.  17.  a^  -  7  a  -f-  12  by  a  -  3. 

14.  a'2_5^.  ^6  by  .X'— 3.  18.  0^  + c  -  12  by  c  + 4. 

15.  a24-7a+ 12  by  «  +  4.  19.  c^  +  c  —  12  by  c  — 3. 

16.  a2  +  7a  +  12  by  a  +  3.  20.  ic2_2ic  — 15  by  a;  —  5. 


118  SPECIAL  PRODUCTS  AND   QUOTIENTS 

DIVIDmG   THE  SUM  OR  DIFFERENCE   OF  TWO  CUBES 
92.     Examples. 


a  +  h\a^+  h^\(fi-ah-\-  b^ 

a  - 

-  b 

\a^- 

-63 

1  a2  +  a6  +  62 

a3  4-  a2  b 

a3 

-a^b 

-a^b 

a^b 

-n^b-ab^ 

a^b- 

-o62 

ab-^+b^ 

ab-^  -  63 

ab-^+b^ 

a62  -  63 

From  these  examples  we  have  the  two  formulas : 
(a3  j^fjz^^^a  +  b)  =  d'-ab  +  b^ 

(a'  -  b')  ^  {a  -  b)  =  a'  +  ab  +  b'- 

That  is, 

The  sum  of  the  cubes  of  tico  ^lumbers  is  divisible  by  the  sum  of 
the  numbers,  and  the  quotient  is  the  sum  of  the  squares  of  the 
numbers  minus  their  product. 

The  difference  of  the  cubes  of  two  numbers  is  divisible  by  the 
difference  of  the  numbers,  and  the  quotient  is  the  sum  of  the  squares 
of  the  numbers  plus  their  product. 

WRITTEN  EXERCISES 

After  performing  the  division  as  indicated,  review  these  ex- 
amples, giving  the  quotients  orally. 

Divide : 

1.  x^  4-?/3  \)j  x-\-y.  10.  a^  —  Sb^hja  —  2b. 

2.  x^  —  y^hyx  —  y.  11.  x^ —  27 y^  hj  x  — oy. 

3.  2^  +  33  by  2 +  3.  12.  1  +  a;H)y  1  +  x^. 

4.  r"'  +  .s'  by  r -\- s.  13.  8  +  a^  by  2  +  a\ 
5.1  +  «-^  by  1  +  a.  14.  1  +  125  x-^  by  1  +  5  x. 

6.  1-b'hyl-b.  15.    2V-1  by  2.r-l. 

7.  a''  +  lbya  +  l.  16.    G4 a^  -  27  ^^  by  4a  - 3  6. 

8.  a^  — Ibya  — 1.  17.    1  — f>m^  hy  1 —  2  m. 


9.    a^ -f  8  ^>M)y  a  4- 2  ?).  18.    l-23mM)yl-2m 


SPECIAL   PRODUCTS   USED   IN  EQUATIONS  119 

USING   SPECIAL   PRODUCTS  IN  EQUATIONS 

Ability  to  form  rapidly  the  special  products  given  by  the 
formulas  in  §§  84-88  is  important  in  solving  certain  kinds  of 
equations.     See  page  86. 

Example.     Solve  {x-\-iy+(x-2y=2(x-{-3)(x-5)+Sl.       (1) 

Solution.     Forming  the  special  products,  we  have 

a;2  -|-2a;+l  -{-  x-  -  Ax  +  4  =  2 x^  - 4:X  -  30  +  SI.  (2) 

Transposing  and  collecting  terms, 

2x  =  -4,  (3) 

x=-2.  (4) 

WRITTEN   EXERCISES 

Solve  the  following  equations : 

1.  x^=(x-3){x  +  6)-12. 

2.  (1  -x){l-  3x)  -  (1  +  xf  =  2a;"  -  18. 

3.  (a  -f  1)  (a  +  2)  +  a  (a  +  3)  =  2a(a  -f  5)  +  2. 

4.  (b-^3)(b-2)-{b  +  iy=-2{b  +  4.)-l. 

5.  (z -  5)2 4-  (^  -  1)  (z  -l)  =  2z(z-S)-  25. 

6.  (a  +  4)2  +(a  -  1)(2  « +  5)  =  (a  +4)(3  a  +  2). 

7.  (a-l)(3a-l)-(a-f  l)2  =  2a2_18. 

8.  (G  -  a)2-f  (a  -  3)(2  a  -  5)  =  (3  a  +  l)(a  -  3)+  84. 

9.  (7  a  -  18)(a  -f  4)-(a  -  1)^=  6(a  +  2)^-  79. 

10.  (2  b-  30)(b  -  1)  -  5  52  =  6  6  -  3(6  +  5)2  +  65. 

11.  (5  -  c)2  +  (7  -  c)2+(9  -  c)2=(c  -  1)(3  c  -  58)-  93. 

12.  (5  c  -  3)(2  +  c)-  4(c  - 1)2  =  (c  -f  l)2-f  54. 

13.  (8-4c)(5-c)  =  (c  +  l)2+(c  +  3)(3c-8)+218. 

14.  (y-  1)2+  4(2/  4- 1)^+ (1  -  2/)(5  2/  +  6)  =  15  2/  -  29. 

1 5 .  x{x  +  3)  +  (a;  4-  l)(x  +  2)  =  2  a;(a;  +  5)  +  2. 

16.  .r2  =  (a;-3)(a;+ ^)-19. 

17.  (5  +  5a;)(3-a.-)4-2(2'  +  l)2+3(a;  +  l)(a;-7)  =  17(a;-f  1). 

18.  (8  +  3  x)(4  -x)  +  {x  -  1)  (X  -  2)  +  2(a;  +  5)^  =  105. 


120  SPECIAL  PRODUCTS  AND   QUOTIENTS 

PROBLEMS 

1.  Find  two  consecutive  integers  whose  squares  differ  by  51. 

2.  Find  two  consecutive  integers  whose  squares  differ  by  97. 

3.  Find  two  consecutive  integers  whose  squares  differ  by  a. 
Show  from  the  form  of  the  equation  obtained  that  a  must  be 
an  odd  integer. 

4.  There  is  a  square  field  such  that  if  each  of  its  dimen- 
sions is  increased  by  5  rods,  its  area  is  increased  625  square 
rods.     How  large  is  the  field  ? 

Suggestion.  If  a  side  of  the  original  field  is  w^  then  its  area  is  ic-,  and 
the  area  of  the  enlarged  field  is  (iw  +  5)2. 

5.  A  rectangle  is  9  feet  longer  than  it  is  wide.  A  square 
whose  side  is  3  feet  longer  than  the  width  of  the  rectangle  is 
equal  to  the  rectangle  in  area.  What  are  the  dimensions  of 
the  rectangle  ? 

6.  A  rectangle  is  16  feet  longer  than  it  is  wide.  If  its  length 
is  increased  by  4  feet  and  its  width  is  decreased  by  3  feet,  its 
area  is  decreased  by  50  square  feet.     What  are  its  dimensions? 

7.  A  field  is  20  rods  longer  than  it  is  wide.  If  its  length  is 
increased  by  8  rods  and  its  width  is  decreased  by  5  rods,  the 
area  is  decreased  by  50  square  rods.  What  are  the  dimensions 
of  the  field  ? 

8.  A  farmer  has  a  plan  for  a  granary  which  is  to  be  15  feet 
longer  than  wide.  He  finds  that  if  the  length  is  decreased  3 
feet  and  the  width  is  increased  2  feet,  the  floor  space  will  be 
increased  by  16  square  feet.     What  are  the  dimensions  ? 

9.  If  the  length  of  a  rectangular  flower  bed  is  decreased 
4  feet  and  its  width  is  decreased  1  foot,  its  area  will  be  de- 
creased by  39  square  feet.  What  are  its  present  dimensions, 
if  its  length  is  8  feet  greater  than  its  width  ? 


REVIEW   QUESTIONS  121 

MISCELLANEOUS   EXERCISES 

Perform  the  indicated  operations : 

1.  (2a-36  +  c)2.  4.  (2a-36)3. 

2.  (o  x  —  y  —  4:  zf.  5.  {ah  —  cdf. 

3.  (3  m  —  4:  n -{- 2  py.  6.  {Ax-oyf, 

7.  [(a2  +  3)-(?>2_2)]2. 

8.  [(a.'^_2)  +  (2/^-3)]l 

9.  (a;2-7x  +  6)--(a:-6). 

10.  (144  m^  -  64  n"^)  --  (12  m^  -  8  w'). 

11.  [(.'c-2/)2-2;2]-(a;_2/4-2;). 

12.  [(a;  +  2/)'-^']-(^  +  2/-2;). 

13.  (125  a;3_  27  2/3^)^(5  ^_32^>^_ 

REVIEW  QUESTIONS 
State  in  words  each  of  the  following  formulas : 

(a  -hby  =  a'-h2ab-\-  b~. 
(a  -by  =  a'--2ab-\-  b\ 
(a  +  b)(a-b)  =  a^-b\ 
(x  +  a)(x  +  b)  =x''  +  (a-\-  b)x  +  ab. 

{a  -\-  b  +  cy=  a"  +  b-  +  c''  +2  ab  -\-2ac  ^2  be. 
(a  ■{-by  =  a'  +  3  d^b  +  3  a/>2  ^  b\ 

(a  -  by  =  a'-3a-b  +  3  ab-  -  b\ 

(a2  _  ^2)  ^  (a  _^  ^)  _  ^  _  ^ 

(a2  _  62)  H-  (a  -  6)  =  a  +  6. 

(a3  _^  ^3)  ^  (a  _|_  6)  3^  a2  _  a6  4. 1,2 

(a3  _  53)  ^  (^  _  ^>)  ^  ^2  _^  fl^  ^  ^2_ 

[jr2  -h(a  +  b)x  +  a6]--(j(r  +0)=  x  +  b 
[jf2  +(a  +  6)jr  +  ab^^{x  +6)=  ^  +  a 


CHAPTER   VII 
FACTORS  OF  ALGEBRAIC   EXPRESSIONS 

93.  Factors  in  Arithmetic  and  Algebra.  Factors  are  of  great 
importance  in  arithmetic.  Thus  from  the  multiplication  table, 
we  know  the  factors  of  such  numbers  as  25,  42,  49,  54,  63,  etc. 
Likewise  in  algebra  the  factors  of  certain  algebraic  expressions 
are  so  important  that  they  must  be  known  at  sight. 

94.  Prime  Expressions.  An  algebraic  expression  is  said  to 
be  prime  if  it  is  divisible  only  by  itself  and  1.  Factors  of  an 
expression,  when  they  cannot  themselves  be  further  factored, 
are  called  'prime  factors. 

Thus,  2,  o,  X,  x-\-  2,  a^  -|-  W,  are  prime  expressions. 

Case  I:  MONOMIAL  FACTORS 

ax  -\-  ay  -\-  az  =  a{x  +/  +  2). 

95.  Common  Monomial  Factors.  If  the  terms  of  a  polynomial 
contain  a  common  monomial  factor,  the  polynomial  may  be 
divided  by  the  monomial,  and  the  quotient  and  the  divisor  are 
factors  of  the  polynomial. 

E.g.  a?-  —  ab  =  a(a  —  b)  ] 

6xy  —  S  x-y  +  4  x^y  =  xy(6  —  3  a:  +  4  a:^), 
x^  +  Sx^y  +  Sx^y'^  +  xy^  =  x(x^-\-Sx-y  +  Sxy^  +  y^). 

Observe  that  factoring  the  various  parts  of  a  polynomial 
does  not  factor  the  polynomial. 

E.(/.     d^  +  ax  +  aJ>  +  by  is  not  factored  by  writing  it 

a{a  +  a:)+  b{a-\-y). 

Removing  a  common  factor  from  the  terms  of  a  polynomial 
is  nothing  more  than  the  application  of  Principle  I. 

122 


MONOMIAL   FACTORS  123 

ORAL  EXERCISES 

Read  the  following  and  give  the  factors  of  each : 

1.  cc^  -\-  x"^  -^  X.  14.  xjf  —  xh/. 

2.  4a2-h3a-fa3.  15.  a-^  +  4  a;^ -t- 5  a;'^  +  3  a:2. 

3.  aW  +  a^h\  16.  %  (ji:'b'' -  12  a''h\ 

4.  3  a;?/  -  4  xhj\  17.  3  a^^V  -  9  aHii'c'. 

5.  Sab^c  —  2a^bc^  18.  5xyz-^25xYz\ 

6.  5  o^y  _  10  x*if.  19.  4  a6  -  12  a^^. 

7.  2  m3/i  -  3  mn\  20.  3  ci^^^  _  9  ^^3^2. 

8.  4  aa;2?/  -  6  a^xy^.  21.  5  ab^  —  10  a?;^ 

9.  4  ^2/2  —  2  rt\?/.  22.  6  a3?>3  -  3  ab. 

10.  6.T?/-3x22/.  23.    7«2^^  +  14a6. 

11.  4  0^32/4  _|.  8  a; y.  24.    S  a'b -\- 24:  a'b. 

12.  8  a;Y  -  12  xy.  25.    9  a.'2^  -  6  .<?/. 

13.  x^y  +  xy\  26.    lOa;^  — 5a.V- 

» 

WRITTEN  EXERCISES 

Factor  the  following  polynomials  : 

1.  1  a'W +  l-i  orb  +  21  ah\  4.    9  z?^  +  21 'yW  -  18  v^^o^^ 

2.  l^a'b-l&a?b^-2a?b\  5.    12  a^ft^  _  g  ^,354  _  g  ^^252^ 

3.  15a;2/'-20aj3y +  a;V.  6.    11  alr^- 44a3a^  +  33ci.r. 

7.  72a6V-36a262x^-48a*5l'c3. 

8.  84  6V2/4  +  18  6V2/4  +  12  63a;y. 

9.  17  a*6V  4- 51  a3?>V- 34  a262c4. 

10.  38  ai25i4c4  _  76  aii5i2c3  -  76  a^^b^'^c'. 

11.  4  a;2«?/ 3«'  +  6  a.-^^^^ft  _  §  aj5a^46^ 

12.  3  a^^b^"  +  6  a<^"6^"  —  12  a^^'b^. 

13.  2  xf'y^  +  4  ar3'»?y46  _  f,  a-^^/Sft^ 

14.  3  £c2«2/2«'  -I-  6  a^Y^  —  9  x-*"?/"'. 

15.  4  x''-"!^  +  6  a^"  2/^  —  9  a^s"?/*^ 

16.  5  x'^'^f  +  15  x^y'^^  —  20  a;^"^/"*. 


124  FACTORING 

Case  II :  TRINOMIAL  SQUARES 

a'  +  2  a6  4-  6^  =  (a  +  6)  (a  +  6),  (1) 

a^-2ab-^b'=(a-b)(a-  b).  (2) 

96.  Square  Root.  If  an  expression  is  the  product  of  two 
equal  factors,  either  of  these  factors  is  called  its  square  root. 

Thus,  a  is  the  square  root  of  a?,  since  a^  =  a  •  a. 
In  §§84  and  85  we  found  by  multiplication: 

(a  +  h)  (a  +  ^)  =  ^2  +  2  a^  +  b% 
and  (a  —  b)(a  —  b)  =  a^  —  2ab  -{-  b^. 

Hence,  a  +  6  is  the  square  root  oi  o?  -\- 2  ab  -\-  b^  and  a  —  6  is 
the  square  root  of  a^  —  2  ab  -f-  b^. 

97.  Trinomial  Squares.  A  trinomial  which  is  the  square  of 
a  binomial  is  called  a  trinomial  square. 

Thus  a^  4-  2  a6  +  6^  and  a-  —  2  ab  -\-  b^  are  trinomial  squares. 

98.  From  a  study  of  the  two  trinomials  a^  -{- 2  ab  +  b"^  and 
a^  —  2  ab  +  6^,  we  learn  to  distingui3h  whether  any  given  tri- 
nomial is  a  perfect  square,  as  in  the  following  examples  : 

1.  cc^  +  4x  +  4  is  in  the  form  of  (1),  since  x^  and  4  are  squares  each 
with  the  sign  +,  and  4  a:  is  twice  the  product  of  the  square  roots  of  x^ 
and  4.     Hence 

a:2  +  4  X  +  4  =  a;--^  +  2(2  x)  +  ^'  =  (x  +  2)  (a;  +  2)  =  (x  +  2)2. 

2.  a:2  —  4x  +  4  is  in  the  form  of  (2),  since  it  differs  from  (1)  only  in 
the  sign  of  the  middle  term.     Thus 

a:2  _  4  X  +  4  =  x2  -  2(2  x)  +  2-2  =  (a:  -  2) (a:  -  2)  =  (x  -  2)2. 

99.  The  foregoing  examples  lead  to  the  following 

Rule.  A  trinomial  is  a  perfect  square  if  it  contains  two 
terms  which  are  squares,  each  with  tJie  si£n  +,  wJiiJe  the 
tJtird  term,  whose  si^n  is  either  -\-  or  —,  is  twice  tJie  prod- 
uct of  the  square  roots  of  the  otiier  two. 

The  square  root  of  such  a  trinomial  is  the  sum  or  the 
difference  of  these  square  roots  a^cordin£  as  the  si£n  of 
the  third  term  is  -\-  or  —. 


TRINOMIAL   SQUARES  125 

ORAL   EXERCISES 

Determine   whether   the    following  are   trinomial    squares, 
and  if  so  indicate  the  two  equal  factors. 

1.  a2  -f-  2  ad  +  d\  10.  a^  -{- b^  -  2  a'^hK 

2.  x^-\-2xy^y\  11.  64  +  ^-  — 16^. 

3.  x^-2xy  +  y\  12.  lG  +  .^•2-8a;. 

4.  a^  +  2  «^/  +  y\  13.  9  —  6  ?/  +  if. 

5.  ci:^-2xY-\-y^'  14.  25  ^2  _^  ;i^g  2/2 -f- 40  a;^/. 

6.  7/1^  -f-  7^2  —  2  mw.  15.  4  m^  H-  n^  +  2  77in. 

7.  r-  +  s''-^2rs.  16.  100 +  52  + 20  s. 

8.  4a;2-8a;?/  +  4/.  17.  64  +  49+112. 

9.  a«  +  6«  +  2  a^ft^.  18.  16  a^  +  25  6^  -  50  a6. 

WRITTEN  EXERCISES 

Decide  which  of  the  following  are  trinomial  squares.     Find 
the  square  roots  of  all  such: 

1.  9 +2.3.4+ 16.  16.  121  +  4  a.-*  -  44  a;^ 

2.  a;2  +  4  2/2  +  4  a;?/.  17.  16a;4  +  64^  -  64a^2/^. 

3.  9a;2_|_i8a;2/  +  9  2/2.  18.  81  a^  -  216  a  +  144. 

4.  4  a;2 -f- 4  a;?/ +  2/2.  19.  4  a2  +  8  rt^2  _|.  4  52^ 

5.  4a;2  +  8a;2/  +  4  2/2.  20.  9  5^  +  18  62c'' +  9  c^. 

6.  25  a?^  + 12  a;?/  +  4  2/2.  21.4  x^  +  4  ?/2  —  8  a.;?/. 

7.  16  «2  +  16  a;2/  +  4  2/^.  22.  9  a2  —  16  a6  +  4  &2. 

8.  9r2  +  36rs  +  25s2.  23.  9  a^  -  24  ^26  _^  16  52. 

9.  16  a;«  +  8  ^y  +  ?/2.  24.  25  +  49  a;2  —  70  x. 

10.  4a^+12aV  +  9a«.  25.  -  30a62  +  9  ^2  +  256^ 

11.  a^o  +  6  a^ft  +  9  62.  26.  16  a2  -  24a6  +  9  52. 

12.  (a  +  1)2  +  2(a  +  1)6  +  52.  27.  36  a.- -  84  x  +  49. 

13.  (a;+3)2+4(a;+3)2/+42/2.  28.  25-90  +  81. 

14.  a;6  +  12  a;3  _|_  36^  29.  64  .^'2  -  32  a;  +  9. 

15.  a^  +  18a2+12.  30.  (3  +  a)2+ 62  -  2  6(3  +  a). 


126  FACTORING 

Case  III :        THE  DIFFERENCE  OF  TWO   SQUARES 

a'^-b^={^a^  b)  (a  -  b). 
100.    In  §  87,  we  found  by  multiplication, 

(a-{-b){a-b)  =  a'-b\ 
Hence  we  have  the  formula 

a'-b'={a  +  b){a-b) 

From  this  formula  we  obtain  the  following 

Rule.  Every  binomial  which  is  the  differejice  between 
two  perfect  squares  is  the  product  of  two  binomial  factors; 
namely,  the  sum  and  the  difference  of  the  square  roots  of 
these  squares. 

E.g.  16  X-  —  9  ?/2  is  the  difference  of  the  two  squares,  (4  x^  and  (3  y^. 
Hence  we  have 

16x2  —  9  2/2  =  (4x)2-  (3?/)2=  (4a:  +  3?/)(4x-32/). 

ORAL  EXERCISES 

In  each  of  the  following  expressions  determine  whether  it  is 
the  difference  of  two  squares,  and  if  so,  find  the  factors. 

27.  {x  -h  3)2 -25. 

28.  25  -  (a  -h  h)\ 

29.  36  -  4(a  -h  h)\ 

30.  9(a  -  by  -  4. 

31.  16(a-6)2-4c2. 

32.  16(a-Z>)2-9d2. 

33.  4  -  9(a  +  by. 

34.  9  -  16(a  +  by 

35.  9  a2  —  4(a;  -f  Tj'y. 

36.  25(.f-y)2-422. 

37.  36(.«4-?/)2— 252*. 

38.  (;4.r2-49(a4-?>)2. 

39.  81a^-36(6H-c)2. 


1. 

a;2  —  4  i/2. 

14. 

l-(x-^yy. 

2. 

9a;2-  36  2/2. 

15. 

4-(.T  +  2  7/)2. 

3. 

x''  -  b\ 

16. 

16  0?  -  25  b\ 

4. 

4a;2-968. 

17. 

49  x^  -  4  2/2. 

5. 

16  a^  -  9  b\ 

18. 

225  -  64  xY- 

6. 

64  -  ly: 

19. 

81a2-144?/2. 

7. 

1  -  b\ 

20. 

58  -  3«. 

8. 

a2-  1. 

21. 

x'  -  81 2/2. 

9. 

1  -9a^. 

22. 

a}  -  (x  +  yy. 

10. 

4-36  a''. 

23. 

{x  +  2/)2  -  a\ 

11. 

1  -  r)4  a\ 

24. 

(,,  _  yy  _  a2. 

12. 

144  x^h"  -  1. 

25. 

a2_(x-2/)2. 

13. 

36  a'b^  -  c\ 

26. 

(a  4.  3)2  -  16. 

THE   DIFFERENCE   OF  TWO   SQUARES  127 

101.  Following  is  another  example  of  an  expression  which 
may  be  written  as  the  difference  of  two  squares : 

E.g.     (fi -\- h'^  +  2  ah  -  c-  =(«  +  b)^  -  c-^  =(«  +  6  +  c)(a  +  6-c). 

WRITTEN   EXERCISES 

Factor  each  of  the  following : 

1.  x''-{y-zy.  9.  (3a-2hy-{8a  +  5by. 

2.  (x-yy-z\  10.  (3  771-4)2 -(2  m +  3)2. 

3.  a2_|_^2_2rt/>-4.  11.  (2r  +  sy-(3r-sy. 

4.  x''-{-y''-2xy-z\  12.  81  -  (a  +  6 -f  c)2. 

5.  4a262-(«2_f-&2_^2^)2^  13_  x4^2x-  +  l-4:a-. 

6.  a2-(62  +  c2  +  2  6c).  14.    a'' -  (x -\- 2  yy. 

7.  (2  a -5)2 -(3  a +  1)2.  15.    9a;2-(a-6)2. 

8.  {3x^-7jy-(x-^yy.  16.    25m2-(3r  +  2.s)2. 

102.  Expressions  Reducible  to  the  Difference  of  Two  Squares.* 

Example.  The  trinomial  a"*  +  a262  +  6"*  would  be  the  difference 
of  two  squares  if  its  middle  term  were  2  a262  instead  of  a^b^- 
Hence,  if  we  add  a^b^  to  this  term  and  subtract  aW  from  the 
whole  expression,  we  shall  have  the  difference  of  two  squares. 

Thus,  a4  4.  a'^i)2  +  ^^4  ^  «4  _,_  o  a^b^  +  &*  -  a^5-'  =  {a~  +  b'^y-  -  a^-b'\ 

WRITTEN  EXERCISES 

Factor  the  following : 

1.  ar*  +  a;2?/2  +  2/4.  8.  x^  —  14:  xhf  +  ^o  y*. 

2.  x^-\-x*y^  +  y\  9.  4.  a' -  29  aW- -^  25  b\ 

3.  a'  +  4  6*  =  (a^  +  4  a'b'^  +  4  6^)  -  4  a262. 

4.  m8  +  4?i.8.  10.  16  a^  +  20  «2^2  ^  9  ^4^ 

5.  a4  +  a2  +  l.  11.  cc*"  +  a2'»62"  +  6-*". 

6.  /  + 2/^  +  1.  12.  ic^  -  12 .1-2/ +  4  ?/4. 

7.  4:x'  +  llxY-\-^y*-  13.  a'-lTaVf~-^16b\ 

*  This  article  may  be  omitted  without  destroying  the  continuity. 


128  FACTORING 

Case  IV :  the  sum  OF  TWO   CUBES 

a'  -^  b'  =  {a+  b){a''  -  ab  +  b'). 

103.    In  §  92  we  found 

(a'  +  b')  -^(a-\-b)  =  a2  ^  ab -\-  b\ 

Since  Dividend  =  Divisor  x  Quotient,  we  have 
a^  +  6^  =  (a  +  b){a'  -  ab  +  b^). 

From  this  formula  we  obtain  the  following 

Rule.  The  sum  of  the  ciibes  of  two  ninnhers  is  the  product 
of  two  factors,  one  of  which  is  the  sum  of  the  numbers,  and 
the  other  is  the  sum  of  their  squares  minus  their  product. 

E.g.  (1)  :rJ^ -\-y'^  =  {x  +  y){x'^  —  xy +  y'^). 

(2)  8  a3  +  27  h^  =  (2  ay  +  (3  &)«. 

=  (2  a  +  3  6)(4  a2  -  2  a  .  3  6  +  9  &2). 

(3)  X^  +  y^=   {X'Y  +  (?/2)3  =   (x2  +  y2)(a;4  _  r^lyl  +  y4). 

Notice  the  difference  between  the  trinomial  x^  —  xy  +  y"^  and 
the  trinomial  square  x"^  —  2xy-\-  y^. 

EXERCISES 

Determine  whether  each  of  the  following  is  the  sum  of  two 
cubes,  and  if  so  find  the  factors.     Read  1-6  at  sight. 

1.  x^  +  y\  10.  8a^-\-27b\  19.  64:  x'- -\- 27  y\ 

2.  a^  +  S/A  11.  8a3  +  64  63.  20.  8^ -|- 101 

3.  27a^-\-b\  12.  tv'x^-\-x^a\  21.  l  +  729x«. 

4.  8cc'-\-l.  13.  l-hSa^^'.  22.  x'-i-y^^ 

5.  l-\-6ix^.  14.  64.7^  +  343.  23.  a^-\-b\ 

6.  23 +  33.  15.  l-h«^  24.  27?-3  +  125.s^ 

7.  125  +  729.  16.  a^  +  9/A  25.  x^-^27y\ 

8.  l  +  125ic^  17.  125  .r' -h  ?/6.  26.  64  +  a^ 

9.  27  a;^  4-1.  18.  l4-a.-8.  27.  a36«-|af»?A 
28.    Find  whether  a^  +  ?/3  is  exactly  divisible  by  x  —  y. 


THE   DIFFERENCE   OF  TWO   CUBES  129 

Case  V :  THE  DIFFERENCE  OF  TWO  CUBES 

a'  -  b'  =  (a  -  6)(a2  -\- ab -{-  b^). 

104.   In  §  92  we  found 

(rt3  _  ^3)  ^  (^^  -b)  =  a2  +  ah  -f  b-. 

Since  Dividend  =  Divisor  x  Quotient,  we  have 
a'-b'  =  {a-  b){a'  4-  fl/>  4-  62). 

From  this  formula  we  obtain  the  following 

Rule.  The  difference  of  the  cubes  of  two  numbers  is  tJie 
product  of  two  factors,  one  of  which  is  the  difference  of 
the  numbers,  and  the  other  is  the  sum,  of  their  squares 
plus  their  product. 

E.g.  (1)  x3  -  ?/3  ={x-y) {x^  +  xy  +  y^). 

(2)  8a3_64  53=(2a)3_(4&)=^ 

=  (2  a  -  4  6)(4  a2  +  2  a  .  4  6  +  16  62). 

(3)  a"^  -  W  ={a^)^-{h^y  =  {a'^  -  h'^){a''  +  d^h^  -{-  b*^). 

Notice  the  difference  between  the  factor  x^  -\-  xy  +  ?/,  and  the 
trinomial  square  x^  +  2  xy  +  y"^. 

EXERCISES 

Determine  whether  each  of  the  following  is  the  difference 
of  two  cubes,  and  if  so,  find  the  factors.     Read  1-6  at  sight. 


1. 

J.3  _  ^3^ 

8. 

1-8  a\ 

15. 

27  .^•3  -  64. 

2. 

x^-1. 

9. 

04  a^-?/. 

16. 

'2\r'  - 1. 

3. 

l-x\ 

10. 

27  - 125  a\ 

17. 

8^-/- 

4. 

x^  —  y^. 

11. 

a?  —  y^. 

18. 

64  a^  -  27  b\ 

5. 

1-f. 

12. 

x^  -  8. 

19. 

l-729.r«. 

6. 

f-1. 

13. 

l-125.rl 

20. 

.f6  -  ?/i^ 

7. 

1  -  rt^o. 

14. 

8-27.^3. 

21. 

27  i""  -  125  s". 

22.    Also  factor  Examples  4,  5,  6, 19,  and  20  as  the  difference 
of  two  squares ;  and  then  resolve  these  factors  still  further. 


130  FACTORING 

Case  VI :      TRINOMIALS   OF  THE  FORM  X^  +  px  +  q 

105.  In  §  88  were  found  such  products  as 

(1)  (x-{-5){x-j-2)=x''-{-7x-^10. 

(2)  (x  -  5)  (x  --2)  =  a;2  _  7  X  +  10. 

(3)  (a.'  +  5)(.r-2)=.r2-h3a.'-10. 

(4)  {x-b)(x-{-2)=x''-^x-10. 

All  these  are  included  in  the  form 

(jr  -h  a)(jr  +  6)=  jr^  +(a  +b)x  +  ab, 

in  which  the  coefficient  of  x  is  the  algebraic  sum  of  a  and  b  and 
the  last  term  is  their  product. 

106.  It  is  possible  to  determine  at  sight  whether  a  trinomial 
is  of  the  form  just  considered,  and  if  it  is,  to  find  the  factors  by- 
inspection. 

Illustrative  Examples.  Determine  whether  the  following  tri- 
nomials can  be  factored  by  inspection : 

1.  a;2  +  7  ic  4-  12.  The  question  is  whether  two  numbers  can  be  found 
such  that  their  sum  is  +  7  and  their  product  12.  3  and  4  are  such  num- 
bers.    Hence,  ^2  +  7  .^  ^  12  ^z  (x  +  3)  (a:  +  4) . 

2.  x^  —  ^x—  14.  Since  the  product  of  the  numbers  sought  is  —  14, 
one  number  must  have  the  sign  —  and  the  other  +  ;  and  since  their  sum 
is  —  5,  the  one  having  the  greater  absolute  value  must  ha  .'3  the  sign  — . 
The  numbers  are  —  7  and  +  2,  and  we  have  a:-  — 5a:— 14  =^x—7)(;x  +  2). 

3.  a:2  -  7  X  +  12  =  (x  -  3)(x  -  4).       Since    (-  3) (-  4)  =  -\-  12    and 

(_3)  +  (-4)  =  -7. 

4.  S-2  + 4x  -  12  =(x  + 6)(a-- 2).  Since  (+6)(-2)  =  -12  and 
(+6)  +  (-2)  =  +  4. 

These  examples  lead  to  the  following 

Rule.  To  factor  a  trinomial  such  as  x'-{^px-\-q,  we 
try  to  find  two  numbers,  a  and  b,  whose  product  is  q  and 
ivhose  algebraic  sum  is  p.  If  two  such  numbers  can 
be  found,  then  the  factors  are  x  +  a  and  x  +  b. 


TRINOMIALS  OF  THE  FORM  x^-^-px-l-q              131 
ORAL  EXERCISES 

In  each  of  the  following  state  what  is  the  product  of  the  two 
numbers  to  be  found  and  what  is  their  sum.  Then  give  the 
factors. 

1.  x^ -{- 3  X -\- 2.  14.   x^  —  4:X—o. 

2.  .1-2-3x4-2.  15.   x''-i-4:X-5. 

3.  x^—x—2.  16.    x^—6x-\-o. 

4.  x^  +  x-2.  17.    .i-2  +  7  a; -f- 6. 

5.  x\-i-  4  a;  +  3.  18.    x"^  -  5  x  —  6. 

6.  x''-2x-o.  19.    x''-\-ux-6. 
7o   x^  +  2x-  3.  20.    .t2  -  7  X  +  6. 

8.  x'^-Ax  +  S.  21.    .r2  +  5a;  +  6. 

9.  a;2  +5  a:  +  4.  22.    .v^  _  a;  —  6. 

10.  x2—  5  a;  -h  4.  23.    .t^  +  a;  —  6. 

11.  x--3a;-4.  24.    a.'2-5a;  +  6. 

12.  X-  -]-3x—4:.  25.    :r2  +  7  a^  +  12. 

13.  X-  -\-6x+  5.  26.    X-  ~x  —  12. 

Give  orally  or  in  writing  the  factors  of  the  following : 

27.  a;2-f  a.-~12.  40.    a.-^  -  10  a;  +  9. 

28.  a:2-7aj+12.  41.    a;2  +  8a;-9. 

29.  x^-{-6x-\-S.  42.    a;2-8a;-9. 

30.  .^'2  -  2  a;  -  8.  43.    x-  +  7  .^-  +  10. 

31.  a^H-2a;-8.  44.    a;2-7a;  +  10. 

32.  a.-2-6a;-f  8.  45.    x-  +  3x--10. 

33.  a;2  +  8a;  +  7.  46.    .r2-3.r-10. 

34.  a;2  _  8  aj  +  7.  47.    x-  +  8  a;  +  12. 

35.  x''-6x-7.  48.    a-2-8a;4-12. 

36.  a.-2-f6a;-7.  49.    a'2-4.r-12. 

37.  ar^  +  6  x  -f  9.  50.    x-  +  4  a;  -  12. 

38.  a;2-6a?  +  9.  51.    a-2  +  9  a;  +  20. 

39.  a^-^-f  10a;  +  9.  52.    x^-9x+20. 


132  FACTORING 

107.  It  is  not  always  possible  to  factor  by  inspection  expres- 
sions of  the  form  x^  +pa;  +  g;  for  it  may  be  that  there  are  no 
integers  whose  product  is  q  and  whose  algebraic  sum  is  p. 

E.g.  Given  a:-^  +  5  x  +  3.  It  is  easily  seen  that  there  are  no  two  in- 
tegers such  that  their  sum  is  +  5  and  their  product  +  3. 

WRITTEN    EXERCISES 

Determine  whether  each  of  the  following  trinomials  can  be 
factored  by  inspection,  and  if  so,  find  the  factors. 

24.  a' -11  a? +  2^, 

25.  a^  -  11  «2  _  60, 

26.  a^  — 14  a  — 51. 

27.  a?  —  ^a  —  54. 

28.  a^  -  8  a;2  _  32. 

29.  a«-3a3-154. 

30.  a;2-10a;  +  25. 

31.  a?h^  -  13  a¥  -  30. 

32.  ic2  —  17  xyz  +  72  2/V. 

33.  r^  4-  6  r^s  —  91  s^. 

34.  aV  +  9  a^c*  -  162. 

35.  a2+lla-210. 

36.  m^  +  4  mhi  +  4  n^. 

37.  sH"^  -  15  St  -  54. 

38.  a262  _  27  a6  +  26. 

39.  ^2  +  13  Z  -f  42. 

40.  xHf  -  \lxy  -  180. 

41.  9  (r  + 24  a +16. 

42.  81ci2-99aH-30. 

43.  (f  +  26  g  +  133. 

44.  x^  +  5  x^  —  84  f. 

45.  7-2 -f  3  r  -  154. 

46.  2^2  -f  38  wy  +  165  v*. 


1. 

y?^-llx-\-2^. 

2. 

x"-  +  2  a: -35. 

3. 

aj2  _  3  X  -  40. 

4. 

x^-2x~  24. 

5. 

x^-\-x-  30. 

6. 

a;2_o^_  3 

7. 

a;2  _^  2  a.'  -  24. 

8. 

a2  _  4  a  -  32. 

9. 

rt2  4.  4  «  _  32. 

10. 

62+15  6+56. 

11. 

62  +  8  6  + 15. 

12. 

62  _  /j  _  56. 

13. 

62  +  6  -  56. 

14. 

C.2  _  3  c  -  15. 

15. 

a;2  _  15  a;  4.  56. 

16. 

^2+  15  a;  —  54. 

17. 

x^-\^x-m. 

18. 

2/2  +  21 2/ +  98. 

19. 

2/'  -  7  2/  -  98. 

20. 

a.-^  -  19  .X-  +  78. 

21. 

:xA  +  18  a;2  +  77. 

22. 

x^  -  5  a;2  _  104. 

23. 

a2  +  32  a  +  240, 

TRINOMIALS   OF   THE   FORM   ax^ -{- bx -\- c 


133 


Case  VII  :     TRINOMIALS   OF   THE  FORM  OJT^  -\-  bx  +  C. 

108.    Examples. 


(1)  2x  +  5 
Sx  +  2 
6  x^  +  15  x 

4  X  4- 10 

6  x2  +  19  X  +  10 


(2) 


2x  +  5 

3x  -  2 

6  x^  +  15  X 

-4x- 

10 

6  x2  +  11  X  -  10 


In  Example  (1),  the  products  3x  '2x  —  6  x~  and  2-5  =  10 
are  called  end  products  and-  2  •  2  x  =  4:X  and  5  •  3  it*  =  15  x  are 
called  cross  products.  Likewise,  in  Example  (2),  6  x"^  and  —  10 
are  end  products  and  —Ax  and  15  x  are  cross  products. 

In  each  case  we  see  that  the  final  result  is  a  trinomial,  two  of 
whose  terms  are  the  end  products  ivhile  the  third  term  is  the  alge- 
braic sum  of  the  cross  products. 

Likewise,  examine  the  following : 


(3) 


2x-5 

3x  +  2 

6  x2  -  15  X 

4x- 

10 

6x2-  11  X- 

10 

(4) 


2x- 

-5 

3x- 

-2 

6x2. 

-  15x 

-4x 

+  10 

0x2 

-  19x 

+  10 

WRITTEN  EXERCISES 

In  this  manner  obtain  the  following  products: 


1. 

(2a  +  3)(a  +  3). 

12.    (. 

2. 

(4  a  -  1)(3  a  +  2). 

13.    (, 

3. 

(2a;+5)(a;-7). 

14.    (' 

4. 

(7r  +  8)(3r-G). 

15.    ( 

5. 

{2x-\-S)(9x-4:). 

16.    (- 

6. 

(3m-l)(4m  +  3). 

17.    ( 

7. 

(5.s-7)(2s-4). 

18.    ( 

8. 

(2a;-l)(7a;  +  4). 

19.    ( 

9. 

(4?i-  9)(5n-  7). 

20.    ( 

10. 

(8  7/-l)(5yH-ll). 

21.    ( 

11. 

{t-5){t  +  A). 

22.    ( 

5  X  -  y)(2  X  -  3  y). 
3x-2y){x-\-3y). 
■i  a  -  3  2j){a -{- y). 
3r-2s){2r-\-s). 
5  ni  —  n)(2  m  +  n). 
5aH-3.i')(3a— 4  a;). 
4a-5?>)(a  +  3  6). 
3a-\-5b)(a-b). 
3c-  Td)(2c  +  3d). 
2a-3b){3a-^2b). 
6x-5y){2x-\-3y). 


134  FACTORING 

109.  Factoring  ax"^  -\-  bx  -\-  c  hy  Inspection.  Trinomials  in  the 
form  ax^  -^-hx  -\-  c  may  sometimes  be  factored  by  inspection. 

Example  1.     Factor  5  ic^  +  16  a;  +  3. 

If  this  is  the  product  of  two  binomials  they  must  be  such  that  the  end 
products  are  5  x^  and  3  and  the  sum  of  the  cross  products  16  x. 

One  pair  of  binomials  having  the  required  end  products  is  5x  +  3  and 
o:  +  1.  Others  are  5 x  +  1  and  a:  +  3 ;  bx—\  and  a;  —  3 ;  and  5 x  —  3  and 
x-l. 

It  is  convenient  to  write  dovm  these  possible  pairs  of  factors  as  follows, 
as  if  arranged  for  multiplication  : 

5a:  +  3  5a:  — 3  Scc  +  l  5a;— 1 

x  +  1  x  —  l  a;  +  3  x—  3 

The  sum  of  the  cross  products  in  the  first  pair  is  8  x,  in  the  second 
pair  —  8  a-,  in  the  third  pair  16  x,  and  in  the  fourth  —  16  x.  Since  16  x 
is  the  middle  term  required,  the  factors  are  5  x  +  1  and  x  +  3. 

Example  2.     Factor  6  a;^  -  19  a;  +  10. 

Pairs  of  binomials  which  give  the  right  end  products  are 

3x  +  5  3x  — 5  2x  +  5  2x  — 5 

2x  +  2  2x-2  3x  +  2  3x-2 

Of  these,  the  ones  which  give  the  right  cross  products  are  2  x  —  5  and 
3x-2. 

Hence  6x2  -  19x  +  10  =  (2  x  -  5)(3x  -  2). 

WRITTEN   EXERCISES 

Factor  the  following : 

1.  2a;2  +  5i»  +  2.      .  5.  2a;2  +  7  a;  +  3. 

2.  2a;2-|-3x-2.  6.  2  a;^  -  7  a; -f- 3. 

3.  2a.'2-3a;-2.  7.  2x^-\-5x-S. 

4.  2a;2-5a;  +  2.  8.  2a;2-5a;-3. 

From  these  examples  we  deduce  the  following 

Rule.  To  factor  a  trinomial  of  the  form  ax"-  -\-  bx  -\-  c, 
ivrite  down  the  ])0fisiMe  pairs  of  binomiaJs  which  £ive  the 
j/j'oper  end  products.  Select  that  pair  whose  cross  prod- 
ucts give  the  proper  algebraic  sum. 


TRINOMIALS   OF  THE   FORM   ax^ -{- bx -\- c              135 
WRITTEN  EXERCISES 

In  this  manner  factor  the  following: 

1.  3»2  +  5a;  +  2.  14.    5  x'' -  6  x -h  1. 

2.  3  a;2  -{.  x  -  2.  15.    5  a;^  +  4  a;  -  1. 

3.  3cc2  — 5.X-  +  2.  16.    oa;2—  4a.'  — 1. 

4.  3  .^2  -x-2.  17.    ()  a;2  -h  7  X'  +  1. 
6.    3ic2_^7a;  +  2.  18.    Ga;"-  — 7a;  +  l. 

6.  3a;2-7x  +  2.  19.    3a;-  +  4a^+l. 

7.  3a;2-h  5a; -2.  20.    3.1-2- 4a' +  1. 

8.  3  a;2  -  5  a;  —  2.  21.    oa;^  —  17  a;  -  12. 

9.  3  a;2  +  17  x  +  10.  22.    T)  x"-  +  17  a;  —  12. 

10.  3a;2-17.a;  +  10.  23.    9  a^  +  9  a  +  2. 

11.  3a;2  +  13a;-10.  24.    2a-2  +  11  a;+ 12. 

12.  3.^2- 13a;- 10.  25.    9  a:^  +  36  a; -f  32. 

13.  5a;2-h6.v  +  l-  26.    2a;2-a;-28. 

In  the  following  try  to  find  the  factors  without  writing  all 

the  pairs  which  give  proper  end  products. 

27.  12s2  +  ll.s  +  2.  39.    3a2_21a  +  30. 

28.  or- +  7^ -3.  40.    6^2^  4^  _  2. 

29.  6a;2-a;-2.  41.    20  a2  -  a  -  99. 

30.  5  7-2  +  18  r- 8.  42.    12  c2 -j- 25  c -h  12. 

31.  14  a2- 39  a +  10.  43.   8  +  6a-5a2. 

32.  5a'2  +  26a;-24.  44.    15  -  5  .t  -  10  a.'2. 

33.  2a;2-5a;  +  2.  45.    6-5a;-4a;2. 

34.  2i)f--m-3.  46.    3/^2-13/^  +  14. 

35.  7c2-3c-4.  47.    15r2-r-2. 

36.  5.i'*  +  9a'2  — 18.  48.    2t''  +  llt  +  D. 

37.  7a4  +  123a2_54  49.    10  -  5.r- 15a;2. 

38.  6  c2- 19c +  15.  50.   5.i'2-33a;  +  18. 


136  FACTORING 

Case  VIII :         FACTORS  FOUND  BY  GROUPING 

ax  +  ay  +  bx  +  by  =  {a  +  b){x  +/). 

110.  Another  method  of  general  application  will  now  be 
applied  to  polynomials  of  four  terms. 

Example  1.     Find  the  factors  of  ax  +  a/y  +  hx  +  by. 

By  Principle  I,  the  first  two  terms  may  be  added  and  also  the  last  two. 
Thus,  ax  +  ay  +  hx  -{-hy  =  a{x  +  y)-\-  b(x  +  y). 

These  two  compound  terms  have  a  common  factor^  (^  -H  2/)?  ^^^  i^3,y 
be  added  with  respect  to  this  factor  by  Principle  I. 
Thus,  a{x  +  y)+h{x-\-y)={a-\-h){x  +  y). 

Hence,  ax  + ay  +  hx -\- by  =  {a +  h){x  +  y). 

Example  2.  Factor  ax  —  ay  —  hx  -f  by. 

Combining  the  first  two  terms  with  respect  to  a  and  the  second  two 
with  respect  to  —  6,  we  have, 

ax  —  ay  —  hx  +  hy  =  a(x  —  y)—  h(x  —  y). 
Again  combining  with  respect  to  the  factor  x  —  y, 
ax  —  ay  —  hx  +  hy  =  (a  —  b){x  —  y). 

The  success  of  this  method  depends  upon  the  possibility  of 
so  grouping  and  combining  the  terms  as  to  reveal  a  common 
binomial  factor. 

WRITTEN   EXERCISES 

Factor  the  following : 

1 .  ab'^-\-  ac^  -  db^  -  dc\  11.  2  n^  -en +2  nd  -  cd. 

2.  6  ms  — 15  nt-{-9ns— 10  mt.  12.  5  ax  —  15  ay  —  3  6.v  +  9  by. 

3.  Sax  —  10ay  +  4.bx  —  5by.  13.  3a;a  —  12a.-c— a  +  4c. 

4.  2  a^ H- 3  aA: - 14  an  —  21  nk.  14.  3  xy  —  4  mn  -\-  6  my  —  2 xn. 

5.  ac -{- be  +  ad -\-  bd.  15.  7  mn  -f  7  mr  —  2n-  —  2  nr. 

6.  aa;2  —  bx^  -  ay"  +  by"^.  16.  a  —  1  +  a^  —  al 

7.  %ac-20ad-Q>bc-\-15bd.  17.  3,s  +  2  +  G.s^  + l.sl 

8.  2ax—Qbx  +  ^  by  —  ay.  1 8.  a.s^  -  3  bst  —  ast  +  3  W^. 

9.  5 -1-4 a— 15  c- 12 at".  19.  3 ??i?i -f- 6  m^  —  2 am  -  «7i. 
10.  15&-6-20^;c-f  8c.  20.  2ar  ^  2afi -\-2hr -{-2bs. 


THE   SQUARE   OF   A   TRINOMIAL  137 

Case  IX :  the  square  of  a  trinomial 

a"  +  b""  -\-  c^  +  2  ab  +  2  ac  +  2  be  =  {a  +  b  -[-  cf. 
111.    In  §  89  we  found 

(1)  {a  +  h  +  cf=a}  +  }?  +.c2  +  2  a6  +  2  ac  +  2  ftc. 

(2)  (rt  4-  ?>  -  c)2  =  a2  4-  62  +  c'  4-  2  a6  -  2  ac  -  2  he. 

(3)  (a  -  6  +  c)2  =  a2  -f  ?y2  _^  c2  -  2  ah  +  2  ac  -  2  6c. 

(4)  {a-h-  cf  =  a'  4-  62  -f-  c'  -  2  ah  -  2  ac  +  2  he. 

A  study  of  these  forms  enables  us  to  determine  whether  a 
polynomial  of  six  terms  is  a  perfect  square ;  namely, 

(1)  Tliree  of  the  terms  must  he  squares  each  with  the  sign  -\-. 

(2)  Each  of  the  other  three  terms  must  he  twice  the  prodrtct  of 
the  square  roots  of  two  of  the  square  terms. 

(3)  The  signs  of  these  products  must  all  he  +,  or  else  tivo  of 
them  must  he  —  and  one  -{-. 

Example.     Find  whether  the  following  is  a  perfect  square : 

4  a;2  _  12  xy  -l^xz-\-^  y""  +  24  ?/z  +  16  z'^ 

Solution.     The  terms  4  x'^.,  9  ^2^  and  16^2  ^.vq  all  squares,  each  with  the 
sign  +.     The  square  roots  of  these  are  2  ic  or  —  2  .r,  3  y  or  —  3y,  4  2:  or 

—  4  0.     By  trial  we  find   that  2(2  a;)(- 3?/)  =  — 12  a;^/;  2(2x)(-4  0)  = 

—  IQxz;  and2(— 3?/)(— 4  0)  =  24y2;. 

Hence  the  given  polynomial  is  equal  to  (2  x  —  3  y  —  4  2)2. 

WRITTEN   EXERCISES 

In  each  of  the  following  determine  whether  the  polynomial 
is  a  perfect  square,  and  if  so  indicate  its  square  root. 

1.  x'^  +  y'^-\-z'^—2xy  +  2xz  —  2yz. 

2.  a2-8a6  +  1652-2rtc+c2^8^^c_ 

3.  9  .x-2  -\-  4: y^  -\-  z'^  —  12  xy  +  6  xz  —  4: yz. 

4.  7f  —  4?/2  —  8 a.7/2  -f- 16 it-  +  16 x"^  +  4. 

5.  o2  4.  ^^2^2  _  2  a26  +  2  ahc  -  aWc  +  h'^c'^. 

6.  a'6-4a-^  +  4a;4  +  6x-3-12a;2_^9^ 

7.  x"  +  lea.V  +  289  +  ^x'y  +  34a;  +  136x?/. 


138  FACTORING 

Case   X:  THE   REMAINDER   THEOREM* 

112.  It  is  possible  to  tell  whether  a  binomial  like  x  —  2  will 
exactly  divide  a  polynomial  like  x^  —  5x  -\-  4:  without  actually 
performing  the  division. 


Examples.    (1)  x^  —  ^x  +  4 

X-  —  2  X 


x-2  (2)  X-  -  5  X  +  4  'i^ 


x  —  S  X?-  —  X  lic  —  4 


—  3x4-4  — 4x  +  4 

-3x+6  — 4x  +  4 


-2  0 

If  we  substitute  2  for  x  in  x^  —  5  ic  +  4,  we  get  4  —  10  +  4  =  —  2,  which 
is  the  remainder  in  Example  (1). 

If  we  put  1  for  X  in  x?-  —  5  x  -|-  4,  we  get  1  —  5  +  4=0,  which  is  the  re- 
mainder in  Example  (2). 

These  examples  illustrate  the  remainder  theorem. 

Rule  :  If  we  substitute  a  number  k  for  x  in  a  polyno- 
mial involving  x,  tJie  resulting  number  is  the  remainder 
arising  froin  dividing  the  polynojnial  by  x  —  k. 

To  make  this  more  evident,  let  us  divide  a?^  —  5  .f  +  4  by  .y  —  k, 

X-  —  bx  +4   \x—  k 

x^  —  kx  I  X  +  (^-  —  5) 

(A;  — 5)x  +  4 

{k  -  5)x -k'^  +  bk 

k^  —  bk  +  4i.     Remainder. 

We  thus  see  that  this  remainder  is  exactly  like  the  dividend 
with  X  replaced  by  k  according  to  the  rule. 

The  object  of  this  rule  is  to  find  remainders  which  are  zero, 
for  then  the  division  is  exact,  and  the  divisor  is  a  factor  of  the 
given  polynomial. 

Hence  this  rule  is  also  called  the  factor  theorem. 

For  instance,  if  we  put  x  =  2  in  x-  —  5  x  +  4,  we  get  4  —  10  +  4  =  —  2, 
which  is  the  remainder,  according  to  tlie  rule,  when  we  divide  x-—  5x  +  4 
by  X  —  2.     Hence  x  —  2  is  not  a  factor  of  x-  —  b  r  -\-  4. 

But  if  we  put  X  =  1  in  X-  —  5  X  +  4,  the  remainder  is  1—5  +  4  =  0. 
Hence  x  —  1  is  a  factor  of  x^  —  5  x  +  4. 

*  Articles  112-115  may  be  omitted  without  destroying  the  continuity. 


THE   REMAINDER   THEOREM  139 

113.  Finding  Factors  by  the  Remainder  Theorem.  Since  one 
expression  is  divisible  by  anotlier  only  when  the  remainder  is 
zero,  we  use  the  remainder  theorem  to  find  factors  as  in  the 
following  examples : 

Example  1.     Is  «  —  1  a  factor  of  a;-  —  3  i>;  +  2  ? 

Solution.  Substitute  1  for  x  \n  x'^  —  Z  x  -\- 2,  and  we  have  1—3  +  2  =  0. 
Hence  there  is  no  remainder  when  we  divide  x'^  —  Zx  -\-  2\}y  x  —  \.  That 
is,  a;  —  1  is  a  factor  of  ar^  —  3  ic  +  2. 

Example  2.  Show  by  the  remainder  theorem  that  x  —  1  and 
X  -\-'2  are  factors  of  x"^  -{-  x  —  2. 

Solution.     If  we  substitute  1  for  x  in  x^  +  x  —  2,  we  get  1  +  1  —  2=0. 

Hence  there  is  no  remainder  when  we  divide  a;-  +  x  —  2  by  x  —  i. 
That  is,  X  —  1  is  a  factor  of  x-  +  x  —  2. 

To  apply  this  test  to  x  +  2,  we  write  it  in  the  form  x  —  (—  2)  and  then 
substitute  —  2  for  x  in  x"'^  +  x  —  2  and  get  4  —  2  —  2  =  0. 

Hence,  x— (— 2)  =  x  +  2  is  an  exact  divisor  of  x-  +  x  —  2.  That  is, 
X  +  2  is  a  factor  of  x-  +  x  —  2. 

Example  3.     Factor  x'^  +  4  x  —  b  by  the  remainder  theorem. 

We  know  that  if  a  binomial  exactly  divides  x"^  +  4  x  —  5,  its  last  term 
must  be  a  factor  of  5. 

Hence  the  only  possible  binomial  divisors  are  x  —  5,  x  +  5,  x  —  1,  and 
x+  1. 

If  we  substitute  5  for  x  in  x^  +  4  x  —  5,  we  get  25  +  20  —  5  =  40. 

Hence  the  remainder  is  not  zero  when  we  divide  x'-^  +  4  x  —  5  by  x  —  5, 
and  therefore  x  —  5  is  not  a  factor  of  x^  +  4  x  —  5. 

If  we  substitute  —  5  for  x  we  get  25  —  20  —  5  =  0.  Hence  x  —  (—  5)  = 
X  +  5  is  a  factor  of  x''^  -j-  4  x  —  5. 

Likewise,  we  find  that  x  —  1  is  a  factor  and  x  +  1  is  not. 

Hence,  x^  +  4x  —  5=(x  +  5)(x  —  1). 

Example  4.     Is  a^  —  2  a  factor  oi  x^  —  bx^  -\-l  x  —  2? 

By  the  remainder  theorem  the  remainder  is  8—20  +  14—2  =  0.  Hence 
X  —  2  is  a  factor.  To  find  the  other  factor,  we  divide  by  x  —  2  and  get  as 
the  quotient  x'^  —  3  x  +  1. 

Example  5.     Is  .t  —  1  a  factor  of  x~  —  1  ? 

By  the  remainder  theorem,  x  =  1  gives  1"  —  1=1  —  1  =  0.  Hence, 
X  —  1  is  a  factor  of  x^  —  1. 


140  FACTORING 

ORAL  EXERCISES 

1.  Is  cc  —  1  a  factor  of  a.-^  —  1  ? 

2.  Is  ic  —  1  a  factor  of  a^  -f  1  ? 

3.  Is  x-\-l  a  factor  of  a^  —  1  ? 
Suggestion.  Put  a:=— linoc^— 1. 

4.  Is  X  -{-1  a  factor  of  a.-^  +  1  ? 

5.  Is  a;  —  1  a  factor  of  a^^^  —  1  ?  of  a?^  +  1  ? 

6.  Is  a;  +  1  a  factor  of  x^  —  1  ?  of  a;^  +  1  ? 

7.  Is  a?  —  1  a  factor  of  a;''  —  1  ?  of  x^  4- 1  ? 

8.  Is  a;  +  1  a  factor  of  x^  -  1  ?  oiaf-{-l? 

9.  Is  X  -  1  a  factor  of  x^  -  1  ?  of  x'  +  1? 

10.  Is  x  4- 1  a  factor  of  x'  -1?   of  a;^  +  1  ? 

11.  Is  X  +  1  a  factor  of  x^  -  1  ?  of  x^  +  1  ? 

WRITTEN  EXERCISES 

1.  Is  a;  +  1  a  factor  of  .t2  -f-  3  x  +  2  ? 

2.  Is  a?  —  1  or  a; 4- 1  a  factor  of  x"^  —  ^x-}-l? 

3.  Is  X  -  1  a  factor  of  a:^  -  2  ^2  _^  2  x  -  1  ? 

4.  Find  the  factors  of  x^  —  7  x^  +  11  x  —  5. 

5.  Is  X  -  2  a  factor  of  x^  -  8  ?  oi  x^-\-S? 

6.  Is  X  —  y  a  factor  of  x'^  —  y^?    of  xf^  -\-y^?    of  x^  —  y'^?    of 
x6  +  2/5?     ofx^-y^?     ofx^  +  y^? 

Suggestion.     In  each  case  put  y  in  place  of  x  and  see  whether  the  ex- 
pression is  reduced  to  zero. 

7.  lsx-\-y  a  factor  of  x*—y*?      oix*  +  y*?      otx^  —  y^? 
of  x^-{-y^?     of  x^  —  y^  ?     of  x^-\-y^? 

8.  Is  a;  -  1  a  factor  of  x"  -  1  ?    of  x^^  _  ^  9 

9.  Is  a;  +  1  a  factor  of  x"  -  1  ?     of  x^^^-l? 

10.  Factor  a^  —  7  a  -}-  6. 

11.  Factor  a;3  _^  2  ^2  -  a;  —  2. 

12.  Factor  a^  —  a^  —  7  a-  +  a  +  0. 


FACTORS   OF   a;"  +  ?/"  AND  a;"  —  ?/"  141 

FACTORS  OF  jr" +/"  AND   x"  —  y" 

114.  Applying  the  Remainder  Theorem,  By  use  of  the  re- 
mainder theorem  we  may  hiid  under  what  conditions  x-^y  and 
X  ~-y  are  factors  of  x"  +  y"  and  x"  —  2/". 

1.  Is  .T  -f-  ?/  a  factor  of  x'^  +  y"? 

If  we  substitute  —  y  for  a:  we  have  (—  ?/)"  +  ?/".     This  is  zero  only 
when  (—  y)'^  =  — ^";  that  is,  when  n  is  an  odd  integer. 
E.g.  {-yy^=-y^\)w.l  {- vY  = +y''. 

Hence  a;  +  ?/  is  a  factor  of  x^  -f  y^,  but  ?/o^  of  cc^  -f  ?/^- 

2.  Is  a:  +  ?/  a  factor  of  x^  —  2/"  ? 

Here  we  have  (—«/)"  —  ?/".  This  reduces  to  zero  only  when  (— y)" 
—  +  ?/" ;  that  is,  when  n  is  an  even  integer. 

E.g.  (^-y)G  =  ^yQ\)nt{-yy  =-y'. 

Hence  a;  -f  ?/  is  a  factor  of  of  —  y^,  but  not  of  x"  —  2/^. 

3.  Is  a?  —  ?/  a  factor  of  a;"-f  ?/"? 

Since  ?/"  4-  ?/"  is  never  zero,  x  —  y  is  not  a  factor  of  x^  +  y". 
^.^.     X  —  y  is  not  a  factor  of  x^  +  y'^  nor  of  x^  +  y^. 

4.  Is  a?  —  2/  a  factor  of  a.-"  —  2/"  ? 

Since  i/"— 1/"=:0,  ar— ?/  is  a  factor  of  a:"—?/",  for  all  integral  values  of  n. 
E.g.    x  —  y  is  a,  factor  of  x^  —  y^  and  also  of  x^  —  y*. 

Summary.     From  the  foregoing  examples,  we  conclude  that : 

(1)  x-{-y  is  a  factor  of  x"  —  ?/"  ifn  is  even  but  not  if  n  is  odd. 

(2)  x  —  y  is  a  factor  ofx"  —  y"  whether  n  is  even  or  odd. 

(3)  x-\-y  is  a  factor  ofx""  +  ?/'*  ifn  is  odd  but  not  if  n  is  even. 

(4)  x  —  y  is  not  a  factor  of  a."*  +  2/"  *'*^  '"'/  case. 

115.  Special  Case.  When  n  is  an  even  integer,  it  is  best  to 
factor  «"  —  2/"  as  the  difference  of  two  squares. 

E.g.  2c6  —  ?/6  _  (-jcS  _|_  y^')(x^  —  y») 

=  (x  +  y)  (x-  —  xy  +  ?/2) (x  —y) (a;^  -f  xy  +  y-) . 
Also  x^^y^=  (a;4  -  y^){x'^  +  y^) 

=  {x-y)lx  +  y)  (x:^  +  y^)  (x*  +  y^). 


142  J^AUTOKING 

EXERCISES 

1.  Find  one  factor  of  x^  —  y^ ;  also  of  x^^  +  if^. 

2.  Find  two  factors  of  x^^  —  y"^^ ;  also  of  x^^  —  y^^. 

3.  Find  all  the  factors  of  x^  —  y^;  also  of  a^  —  h^. 

4.  Find  all  the  factors  of  x^^  —  y^^ ;  also  of  x^^  —  y^^. 

5.  Find  all  the  factors  of  x^~  —  ?/^l 

6.  Make  a  rule  for  reading  at  sight  the  following  quotients, 
{x^  +y^)  -^  {x-{-y)  and  {x^  —  y^)  -i-  {x  —  y). 

7.  Does    a   similar    rule    apply   to   (x^  -}-  y")  -^  (x-\-  y)    and 
ix'-y-^)^{x-y)? 

8.  Factor  x^  -\-  y^.     Show  that  x"^  -\-y^  is  one  factor  by  sub- 
stituting —  2/2  for  x^  in  (x^y  -f  (y'^y. 

9.  Is  a^  —  2/^  a  factor  of  x^^  —  y^^  ?     Why  ? 

REVIEW   AND   SUMMARY 

1.  What   is   meant   by  factoring  9      Is    x{a -\-h)-\- y{a-\-h) 
factored?     Why? 

2.  By  what  principle  is  the  monomial  factor  removed  from 

ax  -\-  ay  -{-  az  ? 

3.  What  are  the  characteristics  of  the  trinomial  squares : 

a^"  +  2  a"b"  +  6^" ;    a^"  -  2  0^6"  +  6'"  ? 
Are  the  following  trinomials  squares  ?     If  not,  state  why 
they    are    not.      x^  ^xy  ^y^-^     ar*  +  x^y"^  +  2/^5     a^  —  2  a6  —  6^ ; 
4  «2  4_  4  a?>  +  4  62. 

4.  What  are  the  factors  of  the  difference  of  two  squares : 

j(1n  Ji.n  9 

Factor  x^—y^  as  the  difference  of  two  squares 

How  can  x*n  ^  jf2n y2n  _^  yA„ 

be  changed  into  the  difference  of  two  squares  ? 


REVIEW  AND   SUMMARY  143 

5.  What  are  the  factors  of  the  sum  of  two  cubes : 

Can  of  +  y^  be  factored  in  this  way  ?     x^  -f  y^  ? 

6.  What  are  the  factors  of   the  diiference  of  two  cubes: 

Can  x^  —  y^  he  factored  in  this  way  ?  x^  —  y^? 

7.  State  the  conditions  under  which 

x^-i-px  -\-q 
can  be  factored  by  inspection.     Can  a.'^  —  10  a;  -j-  16  be  so  fac- 
tored?    a;2-10a.'- 16? 

8.  Tell  how  to  decide  whether 

fljr^  +  bx  -\-c 

can  be  factored  by  inspection.     Can   6  x"^ -{- IS  xy -\- 6  y"^  be  so 
factored  ? 

9.  What  are  the  factors  of 

ax  -h  a/  -{-  bx  -\-  by? 

Can   x^  +  ax -\- bx -\- ah  be  factored  in  this  way?     Show  that 
x^  —  X  —  3x~  -\-  3  can  be  factored  in  this  way. 

10.    What  are  the  factors  of 

;r2+/  +  z2+  2xy-\-2xz-\-2yz? 
State  the  characteristics  of  the  square  of  a  trinomial. 

11.*  State  the  remainder  theorem.  For  what  values  of  n  is 
X"  +  y"  divisible  by  a;  +  ^  ?  by  x  —  y'!  For  what  values  of  n  is 
it'"  —  2/"  divisible  by  x*  —  ?/  ?  by  x  +  y'! 

Why  is  this  theorem  also  called  the  factor  theorem  ? 

12.  If  li  is  an  integer,  what  kind  of  a  number  is  2  k'!  What 
kind  of  a  number  is  2  Zc  +  1  ? 

13.*    Is  x-y  2i  factor  of  x'^''  -  y^''?  of  x'-^+i  -  //-*+^? 

14.*   Is  a;  +  ?/  a  factor  of  x"-''  +  ^-*?  of  x-"^^  -\-  y-'-^'^? 


144  FACTORING 

MISCELLANEOUS  ORAL  EXERCISES 

Factor  the  following: 

1.  a'b^  +  a^b -\- a\  16.   4:  a""  +  b^ -\- 4t  ab. 

2.  aW-c\  17.    {a-\-by  —  c\ 

3.  ic2  +  4  a?  4-  4.  18.    5  a;2  +  3  a^  +  x*. 

4.  a2-6a  +  9.  19.    5x^-3x^  +  2a^. 

5.  a;2  +  9a;  +  20.  20.   a"  -  IS  a -\- SO, 

6.  a2_5(^^6.  21.   a2  +  13a-f30. 

7.  a;2  +  5  a;  +  6.  22.    a^^  _  (^z  _[_  3)2. 

8.  a;2^11aj  +  30.  23.    (a;-l)2_?/2. 

9.  4  a;2  —  ^z^.  24.    1  +  6  a  +  9  a\ 
10.   a;2  +  7x  +  6.  25.   (x2  +  4a6  +  462. 
11..  a'^  —  25.  26.    a;3H-2/^ 

12.  2ab-\-b'^  +  a^  27.    a;^  -  ^/^ 

13.  a^ -\- 15  a  -  16.  28.    1  —  2/^ 

14.  a;2-lla;  +  30.  29.   y^-1. 

15.  —  2xy  4-  ?/2  +  a;2.  30.   ic^  -f  1. 

MISCELLANEOUS  WRITTEN  EXERCISES 

Classify  the   following   expressions  according  to  the  types 
for  factoring,  and  find  the  factors : 

1.  aj2  -  13  a; -f  42.  10.    9x-  -\- i/ -{-6xif. 

2.  l-8a^.  11.   2y'a^-{-4:ya^-Sya, 

3.  a52  +  17a;+72.  12.    a^- 15  a.- +  36. 

4.  4a;2+9  2/'  +  12a?2/.  13.    9x^ +  S6y' +  S6  xy\ 

5.  4  a;2  +  9 ?/2  _  12  a;?/.  14.    9y —  9z  —  2xy -\-2  xz. 

6.  5  aj2  +  4  eta;  +  7  a;^/-  15.    a' —  1. 

7.  2n^-6nc-Sny-{-9cy.  16.    a"^  +  b* -\- 2 a'b^ 

8.  (a;  +  yy-  (y-2  xf,  17.    a'  +  2  «7>  +  6^  -  c^ 

9.  a?  +  b\  18.   27(x3-125. 


REVIEW  AND   SUMMARY 


145 


19.  4  a2  +  4  a6  +  ¥. 

20.  4  a^  +  9  .'C^  -  12  aic2. 

21.  1+a^. 

22.  2  ^2  +  5  a;  +  3. 

23.  36 -h  4  x^ -f  24  a:^^ 

24.  (a; -1)2-  {x +  iy. 

25.  8+64  a\ 

26.  ac  —  ao;  —  4  6c  -h  4  6x. 

27.  27-216a3. 

28.  ^^-\-Q^a\ 

29.  25(a;  +  l)2-4. 

30.  5  cic  —  10  c  +  4  do;  —  8  d 

31.  4(a;+2)2  +  ?/2-i-4(a.-+2)y. 

32.  ra  +  2rh  —  5  sa  —  10  sh. 

33.  -2a26  +  a^  +  62_ 

34.  2  /ia  —  7^6  +  6  a  —  3  5. 

35.  3(a  +  l)3  +  4(a+l)2  +  «  +  l. 

36.  {x  +  a)2  —  (x  —  ay. 

37.  15  m2  +  34  m  + 15. 

38.  3  ^2  +  27  X  +  42. 

39.  a;4  +  49  a2  +  14  ax\ 

40.  27  a^  —  a^x^. 

41.  33a6  +  aKv\ 

42.  8  6d-40  6e  +  3cd-15ce. 

43.  «2-a;-240. 

44.  (a;  +  2)2-4(.T-2)2. 

45.  X*  -{-9y-  —  6  x^y. 

46.  4a2-7ca2_4d2+7cd2. 

47.  a;2  4- 31  a:  +  240. 

48.  18-27C  +  16  6-24c6. 


49.  4-(a2  +  62-2a6). 

50.  10  ?'  -h  3  6s  —  6  6r  -  5  s. 

51.  25  +  64a;6  +  80a^. 

52.  1000 -cc^. 

53.  lO^+x-s. 

54.  8  a^  +  a3^3  _^  62a2. 

55.  100- 49  a;^ 

56.  100  +  625  +  500. 

57.  a2-17a  +  72. 

58.  a2  +  17  «  +  72. 

59.  a2  +  16  62-8a6. 

60.  x^  —  y^, 

61.  4a2  +  23a-72. 

62.  a;4  + 15x2 -100. 

63.  93  +  81 

64.  9a;^  +  162/''  +  24ar^2/. 

65.  1-1000-1-10\ 

66.  16  a262  +  24  a6  +  36  63. 

67.  64  +  8=4^  +  2'. 

68.  16  a252  _|_  9  fji(.i  ^  24  a'hc. 

69.  a2  +  4  62  +  4  a6  —  4  a;2. 

70.  aW  +  &. 

71.  5a.'3  +  10ar'2/2  +  30a.'3?/^ 

72.  16a2c2  +  4c2a^2_^i5^^a._ 

73.  aV  —  ^^ 

74.  a;4-7a;2-120. 

75.  9  0^62  -  12  0^6  +  4  a2. 

76.  8a6  +  27a6^. 

77.  a;^  + 4.^2  +  4  —  3^. 

78.  1-I25a36l 


146  FACTORING 

79.  16  +  16  a&  +  4  a^fe^.  97.  16  x^ +  9  y*-{-2ia^y^-^9. 

80.  64ci3  +  8a26l  98.  /  +  35  ?/ +  300. 

81.  65r2H-8r-l.  99.  b^f-SOy  +  SOO. 

82.  a2- 13  a -140.  100.  39  a;^  -  16  x^  +  1. 

83.  0^^  +  17  cc^  H- 30.  101.  ac  —  bc  +  ad  —  bd. 

84.  25  -  (a^  -  2  a2Z>3 -I- 5«).  102.  625  -  (31  -  4  a^)^ 

85.  36a2-29a5  +  5&l  103.  z^ -\- ya  -  y^z^  -  af 

86.  a''-a-3S0.  104.  a;^  +  2  a;^  +  1 -x^. 

87.  24  aV  +  a«  +  144  c«a2.  105.  60  a;^  +  7  .t?/ -  2/'. 

88.  9a;2  +  4?/4- 12ay-16.  106.  a.- -  20  aj?/ +  75  3/2. 

89.  <S1  +  100  x8  -  180  a;^  107.  a;^- 17  a; -60. 

90.  «4  +  27a2+180.  108.  36  a^b*  +  c'b' -j- 12  a¥c. 

91.  a4  +  3a2-180.  109.  4.  a" +9  b' +12  ab^ -16  a* 

92.  a^-3a2-180.  110.  100  -  (16  x^  4-2/'-8  .i^). 

93.  lU-(a'-\-b^-2a''b).  111.  6  rd -15  re -\-22cd- 00 ce. 

94.  81a26^H-49c2-126a62c.  112.  -  112  aV  + 49  a4+64  c«. 

95.  12  s2- 23  si +  10^2^  113.  x^-2x^  +  x-2. 

96.  36x^-\-12x'y^-\-f. 

114.  4:  x"^ -\- 9  y- -\- z^  —  12  xy  —  A  xz  +  6  yz. 

115.  a262  +  a2c2  ^  ^2^2  _  2  a26c  +  2  ac^^  _  2  aftc^. 

116.  .^'34-2a;2-a;-2.  117.    a^  +  2a;2- 5  a;  +  2. 

118.  4  a^  -  12  a^  H- 4  ac  4- 9  52 +  c2- 6  &c. 

119.  a6  +  10a3  +  9a2  +  25-6a4-30a. 

120.  a;3-3a;2-a;  +  3.  121.   a.-^  4- 3  ar^  -  a;  -  3. 


CHAPTER   VIII 

EQUATIONS   SOLVED   BY  FACTORING 

116.  Quadratic  Equations.  The  equations  solved  in  the  pre- 
ceding chapters  have  all  been  of  the  first  degree^  that  is,  they 
have  involved  the  first  power  of  the  unknown  and  no  higher 
power,  after  all  possible  reductions  have  been  made. 

An  equation  which  involves  the  second  power,  but  no  higher 
power  of  the  unknown,  is  called  a  quadratic  equation. 

E.g.     x^  +  X  :=  30  is  a  quadratic  equation. 

But  a:2  +  5  a:  +  2  =  a:(x  +  1)  is  of  the  first  degree,  since  it  may  be  re- 
duced to  5  X  -I-  2  =  X. 

117.  One  method  of  solving  quadratic  equations  is  now  to  be 
considered,  namely,  the  method  by  factoring. 

Example.     Solve  the  equation  x^  -\-{x  +  1)^  =  61. 

Solution.     By  i^,  x2  +  x2  +  2  x  +  1  =  61.  (1) 

By  i^,  >S',  2  x2  +  2  X  =  60.  (2) 

By  Z)  I  2,  >S'  I  30  x^  +  x  -  30  =  0.  (3) 

Factoring  the  left  member, 

(r.  +  6)(x-5)=0.  (5) 

This  equation  is  satisfied  by  x  =  5  since  (5  +  6)(5  —  5)  =  11  •  0  =  0, 
and  also  by  X  =- 0  since  (- 6  +  6)(- 6  -  5)  =  0  •  (-  11)  =  0 

It  thus  appears  that  equation  (4)  has  two  solutions,  namely,  5  and  —  (i. 

Each  of  these  values  also  satisfies  equation  (1).  Thus,  5^+ (5  + 1)^=61, 
and    (-6)2 +(-(5  +1)-^  =61. 

In  the  above  solution,  equation  (5)  has  one  member  zero  and 
the  other  member  is  in  the  factored  form.  The  solution,  then, 
consists  in  finding  the  values  of  x  which  make  either  factor 
equal  to  zero,  since  we  know  that  if  one  of  two  factors  is  zero, 
then  the  product  is  zero. 

147 


148  EQUATIONS   SOLVED   BY   FACTORING 

ORAL  EXERCISES 

1.  What  is  the  product  of  3  and  zero  ?   of  6  and  zero  ?  of 
10  and  zero  ?  of  275  and  zero  ? 

2.  If  one  of  two  factors  is  zero,  what  is  the  product  ?     Does 
it  matter  what  the  other  factor  is  ? 

3.  What  is  the  value  of  lii^x  —  1)  if  a;  =  1  ? 

4.  What  is  the  value  of  (x— lj(a;— 5)  if  a;=l  ?     Ifcc  =  o? 

5.  lix  =  2,  what  is  the  value  of  {x  -  2){x^  +  4  a;  -  8)  ? 

6.  If  a;  =  -  3,  what  is  the  value  of  (x  +  ^)(x^  -  2  x-  -h  7)  ? 

7.  Find  a  value  of  x  which  makes  (x  —  3) (a;  +  2)  equal  to 
zero.     Does  this  value  of  x  make  both  factors  equal  to  zero  ? 

8.  Find  a  value  of  x  which  satisfies  the  equation 

(x-  7)(ar'4-2x-3)  =0; 

also  one  which  satisfies  (x  -[-  %){qi?  -{-  x  -{-  4)  =  0. 

Suggestion.  Find  a  value  of  x  which  makes  the  first  factor  zero  in 
each  case. 

9.  Find  two  values  of  x  which  satisfy  {x  —  3) (a:  +  4)  =  0, 
also  two  which  satisfy  (x  +  8)(a;  —  3)  =  0. 

10.  Find  two  values  of  x  which  satisfy  5  a-fa;  -f-  7)  =  0.  Does 
a;  =  0  satisfy  this  equation  ? 

11.  Find  two  values  of  x  which  satisfy  (3  a;  —  2X2  x  -f  5)  =  0. 

118.  Rule  for  Solving  Equations  by  Factoring.  The  method  of 
solution  suggested  by  the  foregoing  examples  consists  of  three 
steps : 

(1)  Transform  the  equation  so  that  all  terms  are  collected  in  the 
left  member,  with  similar  terms  united,  leaving  the  right  member 
zero. 

(2)  Factor  the  expression  on  the  left. 

(3)  Find  the  values  of  the  unknown,  by  setting  each  factor  in 
turn  equal  to  zero  and  solving. 


EQUATIONS   SOLVED   BY   FACTORING 


149 


ORAL  EXERCISES 

"Find  two  solutions  of  each  of  the  following  equations. 

1.  (x  —  l)(x  —  2)  =  0. 

2.  {x  —  4:)(x  -  o)  =  0. 

3.  (x  -  S)(x  —  T)  =  0. 
4. 
5. 
6. 
7. 
8. 


(x  +  3){x  +  2)  =  0. 
(x  +  l){x  +  2;  =  0. 
(x  +  lyx  -  3)  =  0. 
(a;  +  2)(a;  -  5;  =  0. 
far  ~  3)(ar  +  3)  =  0. 
9.    (x  +  4:)(x  +  oj  =  0. 


10. 

(x  +  o)(x  —  5)  =  0. 

11. 

(x-^)(x  +  r)  =  C). 

12. 

{2x-  l){x-\-T)  =  0. 

13. 

(3x-l)(2x-hl;  =  0 

14. 

(2x-  l/x- -h  12)  =  0. 

15. 

(x-i)(x  +  i)  =  0. 

16. 

(4x-l)(2a;4-r)-0. 

17. 

(oa;-l)(3x-  1;  =  0 

18. 

(2x-3)(3a;-  2)  =  0 

WRITTEN  EXERCISES 

Find  two  solutions  for  each  of  the  following  equations : 


1.  .t2-3x  +  2  =  0. 

2.  ^-2  -i-Tx  =  30. 

3.  a-  -  11  a  =  -  30. 

4.  a'-  +  13  «  =  30. 

5.  3x4-^-'=  20^-72. 

6.  17j:  +  30  =  -  x2-40. 

7.  7a:2^2x  =  30^-21. 

8.  lla'  +  3x-  =  20. 


9.  a2  +  10a  +  8=-3a-34 

10.  a-  -{-3  a  =  10  a  +  18. 

11.  a'-  +  10a  =  -24-  4a. 

12.  2x2-G.r  =  -40H-12x. 

13.  a-2  _  16  =  0. 

14.  x'--l  =  0. 

15.  ^2  ^  X  =  0. 

16.  x^  -{-  x  =  0. 


17.  W-DX  +  x-  =  -2x'-20x-2. 

18.  4;/;- =  2."). 

19.  x-  +  ox  +  4  =  0. 

20.  :r'  -  10  X  +  16  =  0. 

21.  x2  +  12x  +  6=  5x^4. 

22.  2x2-7a:  =  60  +  7x. 

23.  60x  +  4x2  + 144  =  8x. 

24.  18x  =  63  -  x2. 


25.  24.1-2  =  12x  + 12. 

26.  2  X  =  63  -  x\ 

27.  22  .r  +  .>.-2  =  363. 

28.  3^2  +  7x  =  6. 

29.  2^2  =  2 -3x. 

30.  X  -  2  =  -  3  x\ 

31.  x2-10  =  3a;. 


152  EQUATIONS   SOLVED   BY   FACTORING 

PROBLEMS   SOLVED  BY  FACTORING 

In  each  of  the  following  problems  find  all  the  solutions  pos- 
sible for  the  equations  and  then  determine  whether  or  not  each 
solution  has  a  reasonable  interpretation  in  the  problem. 

1.  The  sum  of  the  sides  about  the  right  angle  of  a  right  tri- 
angle is  35  inches,  and  the  hypotenuse  is  25  inches.  Find  the 
sides  of  the  triangle. 

2.  The  sum  of  the  length  and  width  of  a  rectangle  is  17 
rods,  and  the  diagonal  is  13  rods.  Find  the  dimensions  of  the 
rectangle. 

3.  A  room  is  3  feet  longer  than  it  is  wide,  and  the  length 
of  the  diagonal  is  15  feet.     Find  the  dimensions  of  the  room. 

4.  The  length  of  the  molding  around  a  rectangular  room  is 
46  feet,  and  the  diagonal  of  the  room  is  17  feet.  Find  its  di- 
mensions. 

5.  The  longest  rod  that  can  be  placed  flat  on  the  bottom  of 
a  certain  trunk  is  45  inches.  The  trunk  is  9  inches  longer 
than  it  is  wide.     What  are  its  dimensions  ? 

6.  The  floor  space  of  a  rectanglar  room  is  180  square  feet, 
and  the  length  of  the  molding  around  the  room  is  56  feet. 
What  are  the  dimensions  of  the  room  ? 

7.  A  rectangular  field  is  20  rods  longer  than  it  is  wide,  and 
its  area  is  2400  square  rods.     What  are  its  dimensions  ? 

8.  A  ceiling  requires  24  square  yards  of  paper,  and  the  border 
is  20  yards  long.     What  are  the  dimensions  of  the  ceiling  ? 

9.  The  area  of  a  triangle  is  18  square  inches,  and  the  sum 
of  the  base  and  altitude  is  12  inches.    Find  the  base  and  altitude. 

10.  The  altitude  of  a  triangle  is  7  inches  less  than  the  base, 
and  the  area  is  130  square  inches.     Find  the  base  and  altitude. 


1 


Pythagoras  (569-500  b.c).  born  on  the  Island  of  Samos,  was 
the  first  of  the  great  Greek  mathematicians.  He  studied  in  Egypt, 
where  no  doubt  he  learned  the  practical  geometry  of  the  Egyp- 
tians. Later  he  returned  to  Samos  to  teach,  but  soon  migrated 
westward  to  Sicily,  and  finally  settled  in  the  Greek  colony  of 
Croton  in  Southern  Italy. 

Here  Pythagoras  became  the  center  of  a  widespread  and  in- 
fluential organization,  a  sort  of  brotherhood  for  the  moral  educa- 
tion  and  purification  of  the  community. 

Pythagoras  is  famous  for  a  system  of  Philosophy  and  for  his 
studies  in  mathematics.  The  name  mathematics  and  the  name 
philosophy  have  been  ascribed  to  him. 


PROBLEMS  SOLVED  BY  FACTORING        153 

11.  The  sum  of  two  numbers  is  17,  and  the  sum  of  their 
squares  is  145.     Find  the  numbers. 

12.  The  difference  of  two  numbers  is  8,  and  the  sum  of  their 
squares  is  274.     Find  the  numbers. 

13.  The  difference  of  two  numbers  is  13,  and  the  difference 
of  their  squares  is  481.     Find  the  numbers. 

14.  The  sum  of  two  numbers  is  40,  and  the  difference  of 
their  squares  is  320.     Find  the  numbers. 

15.  The  sum  of  two  numbers  is  45,  and  their  product  is 
450.     Find  the  numbers. 

16.  The  difference  of  two  numbers  is  32,  and  their  product 
is  833.     What  are  the  numbers  ? 

17.  An  open  box  is  made  from  a  piece  of  paper  20  inches 
square  by  cutting  out  a  5-inch  square  from  each  corner  and 
turning  up  the  sides.  What  is  the  volume  of  the  box  ?  What 
is  the  volume  if  the  original  square  is  x  inches  on  a  side  ? 

18.  An  open  box  is  made  from  a  square  piece  of  tin  by  cut- 
ting out  a  5-inch  square  from  each  corner  and  turning  up  the 
sides.     How  large  is  the  original  square  if  the 

box  contains  180  cubic  inches  ? 

If  ic  =  length  of  a  side  of  the  tin,  then  the  vokime 
of  the  box  is :  b  {x  —  10)  {x  —  10)  =  180.   (See  the  figure.) 

19.  A  rectangular  piece  of  paper  is  20  inches 
long  and  16  inches  wide.  A  box  is  made  by  cut- 
ting a  3-inch  square  out  of  each  corner  and  turning  up  the 
sides.  What  is  the  volume  of  the  box  ?  What  is  the  volume 
if  the  original  paper  is  x  inches  wide  and  .r  -f-  4  inches  long  ? 

20.  A  rectangular  piece  of  tin  is  4  inches  longer  than  it  is 
wide.  An  open  box  containing  840  cubic  inches  is  made  by 
cutting  a  6-inch  square  from  each  corner  and  turning  up  the 
ends  and  sides.     What  are  the  dimensions  of  the  box  ? 


5 

X-  10      -> 

^ 

1 

<2>l 

-5. 

x~io    U 

5 

b  1 

154 


EQUATIONS   SOLVED   BY   FACTORING 


160 


80 


X 
1 

X                   -x  (l60-2x)                X 

X2 

x" 

1 

160 -2x 
160 -2X 

1 

80 


160 


21.  The  dimensions  of  a  j^icture  inside  the  frame  are  8  by 
10  inches.  Find  the  area  of  the  frame  if  its  width  is  2  inches. 
If  its  width  is  a;  inches. 

22.  The  dimensions  of  a  picture  inside  the  frame  are  12  by 
16  inches.  What  is  the  width  of  the  frame  if  its  area  is  288 
square  inches  ? 

23.  A  farmer  has  a  rectangular  wheat  field  160  rods  long  by 

80  rods  wide.  In  cut- 
ting the  grain,  he  cuts 
a  strip  of  equal  width 
around  the  field.  How 
many  acres  has  he  cut 
when  the  width  of  the 
strip  is  8  rods  ? 

24.    How  wide  is  the 
strip  around  the  field  of  problem  23,  if  it  contains  27i  acres  ? 

25.  In  the  ISTorthwest  a  farmer  using  a  steam  plow  starts 
plowing  around  a  rectangular  field  640  by  320  rods.  If  the 
strip  plowed  the  first  day  lacks  16  square 

rods  of  being  24  acres,  how  wide  is  it  ? 

26.  A  rectangular  piece  of  ground  840 
by  640  feet  is  divided  iuto  4  city  blocks 
by  two  streets  60  feet  wide  running 
through  it  at  right  angles.  How  many 
square  feet  are  contained  in  the  streets  ? 

27.  A  farmer  lays  out  two  roads  through  the  middle  of  his 
farm,  one  running  lengthwise  of  the  farm  and  the  other  cross- 
wise. How  wide  are  the  roads  if  the  farm  is  320  by  240  rods, 
and  the  area  of  the  roads  is  1671  square  rods  ? 


840  feet 


60 

ft. 

21 

1 

i-t) 

CHAPTER   IX 
COMMON    FACTORS    AND    MULTIPLES 

121.  Common  Factors.  If  an  expression  is  a  factor  of  each 
of  two  or  more  expressions,  it  is  said  to  be  a  common  factor  of 
these  expressions. 

Thus,  8  is  a  common  factor  of  16  and  48,  and  12  is  a  common 
factor  of  12,  36,  and  48. 

If  each  of  a  given  set  of  expressions  is  separated  into  prime 
factors,  any  common  factor  which  they  may  have  is  at  once 
apparent. 

Illustrative  Example.     Find  the  common  factors  of 

(1)  10(.x  +  yf{x  -  y)  ;  (2)  d{x  +  y)  (x'  -  y') ;  and 

(3)  W{x-\-y)ix'-f). 

Factoring,  10(x  +  yy^{x  -y)  =2-  5{x-}-  y)(x  +  y){x-y).  (1) 

b{x  +  ?/)(x'2  -  ?/2)  =  5(.r  +  y){x^  y){x  -  y).  (2) 

15(x  -f  y)  {x^  -  y^)  =Z-b(x^y){x-  y)  (x-^  +  xy  +  2/2) .  (3) 

The  common  prime  factors  are  5,  x  4-  y,  and  x  —  y.  Tlie 
highest  common  factor  is  the  product  of  these  common  factors, 
namely,  b{x  +  y){x  —  y). 

122.  Highest  Common  Factor.  The  product  of  all  the  com- 
mon prime  factors  of  two  or  more  expressions  is  called 
their  highest  common  factor.  This  is  usually  abbreviated  to 
H.  C.  F. 

The  name  highest,  instead  of  greatest,  common  factor  is  used  in  algebra 
referring  to  the  number  of  prime  factors  which  enter.  Thus,  x-  is  of 
higher  degree  than  x,  although  if  x  =  ^,  x-  is  not  greater  than  x. 

155 


156  COMMON   FACTORS   AND   MULTIPLES 

ORAL  EXERCISES 

Find  the  H.  C.  F.  of  the  following : 

1.  3,  6,  12.  8.    x-y,  ^  -  y\ 

2.  6,  48,  24.  9.    x^  -y'',:x?-  f. 

3.  a,  a}  H-  a.  10.    x^  —  y'^,a^-{-  y^. 

4.  ah,  ac,  a?.  11.   x"^  —  y^,  x^ —  y^. 

5.  ^a\2a\Q>a.  12.    a-h,  a? -2ab-\-h\ 

6.  x-y^x^-y^.  13.    a^  -  6^^  a^  ^  2a&  +  &^ 

7.  x^ry,x?-^y'^.  14.   a^-f  &^  a2-t-2  a6 +  62. 

WRITTEN  EXERCISES 

Find  the  H.  C.  F.  of  the  following : 

1.  x  —  y,  x^  —  7/,   x^  —  2  xy  -\-  y"^. 

2.  x'-\-2x-\-l,   3  .T  +  6  «2  +  3  a^. 

3.  .T2  +  4a;  +  4,   x'^-6x  —  16. 

4.  x''-Sx-\-16,   x^-hlOx-56. 

5.  a'-b\    o}-2ah  +  h\ 

6.  a:^  +  ?/^j   £c^  —  2/2,    cc2  -|-  2  ic?/  +  y"^. 

7.  x2-7ic  +  12,   aa;-3a-6a;  +  36. 

8.  a2- 13  a 4- 42,   a? -21(5,   a'-a-^O. 

9.  27+2/3,   2/^  +  92/  +  18,   2/^-9. 

10.  ^,2^7  6-30,  62  +  116-42,    62-6-6. 

11.  «3  +  2o2  +  a,    a2  +  a,   a3  +  5a2  +  4a. 

12.  y?-\-y^,    x^ -\~  x^y -\- xy^ -\- y^. 

13.  0.-^  + 3x3+2^2,   x'-\-x^,   a:^  +  7a^  +  6a;2. 

14.  a;2-ll.T  +  30,   xz-5z-^x'^-5x. 

15.  m^  —  71^,    2  a;2??i2  _|_  2  .t2?«?i  +  2  a;2n2. 

16.  .^2-1,    .^•3-l,    x2-13a;  +  12. 

17.  1-64  or'',    l-16a;2,   C) -2  z  -  20 x  +  Sxz. 

18.  1  +  125  a',    l  +  10a  +  25a2,    1  -  25  a2. 


COMMON   MULTIPLES  157 

Find  the  H.  C.  F.  of  the  following: 

19.  ac  — ax -\-Sbc  —  Sbx,    a^  +  27  6^ 

20.  5  c  -  2,   5  ac  +  20  c  -  2  a  -  8. 

21.  4:X*  —  x%    2x^-^x^  —  x'',    2x^  —  3x^  +  x\ 

22.  .3  d^  _  3  rt,    3  a^  -  6  a2  _|_  3  a,   6  a^  _^  9  a^  _  W  a. 

23.  6  cc  —  10  37?/  4-  4  xy^,  18  cc  —  8  .r^/^,   54  x  —  16  .t?/'. 

24.  16x^-9if,    12x''-9xy,    16xy-12y\ 

25.  3Gce~-2oh',    18a2+15a&,    24a/j  +  20?>2. 

26.  3aj5  +  9a^  — 3a;^   o  xY  +  ^^  xy'^—5y^,   7  ax^+21  ax-7  a. 

27.  18a73-57aj2_,_3()^^    9ar'- 15.^2  + 6  a!,    18  ar' -39x^+18  a;. 

COMMON   MULTIPLES 

123.  Multiples.  An  algebraic  expression  is  said  to  be  a 
multiple  of  any  of  its  factors.  In  particular,  any  expression  is 
a  multiple  of  itself  and  of  one. 

Thus,  18  is  a  multiple  of  1,  2,  3,  6,  9,  and  18,  but  not  of  8  or  of  12. 
3  a^x^  is  a  multiple  of  3,  3  x,  3  x-,  etc. 

Since  a  multiple  of  an  expression  is  divisible  by  that  expres- 
sion, it  must  contain  as  a  factor  every  factor  of  that  expression. 

E.g.  108  is  a  multiple  of  54  and  contains  as  factors  all  the  prime  factors 
of  54 ;  namely,  3,  3,  3,  and  2. 

124.  Common  Multiples.  An  expression  is  a  common  multiple 
of  two  or  more  expressions  if  it  is  a  multiple  of  each  of  them. 
The  lowest  common  multiple  of  a  set  of  expressions  is  that  one  of 
their  common  multiples  which  contains  the  smallest  number  of 
prime  factors.  The  lowest  common  multiple  is  usually  abbre- 
viated to  L.  C.  M. 

Thus,  x2  —  y-  is  a  common  multiple  of  x  +  y  and  x  —  y  ;  also  x^*  —  y^  is 
a  common  multiple  of  x  +  y  and  x  —  y ;  but  evidently  x-  —  y^  contains  a 
smaller  number  of  prime  factors  than  x*  —  y^,  and  hence  is  the  loicest 
common  multiple. 

The  process  of  finding  the  lowest  common  multiple  of  a  set 
of  expressions  is  shown  in  the  following  example : 


158  COMMON   FACTORS   AND   MULTIPLES 

Illustrative  Example.     Find  the  lowest  common  multiple  of 

(1)  x'-f-;  {2)x'-\-2xy  +  if',  ^m\{Z)x'-2xy  +  y\ 

Factoring,  x-  —  y^  ={x  -  y)(x -{- y).  (1) 

x^  +  2xy  +  y'^=ix-{-y)(x  +  y).  (2) 

x^-2xy +  y^=(ix-y)ix-y).  (3) 

In  order  that  an  expression  may  be  a  multiple  of  (1)  it  must  contain  tlie 

factors  x  —  y  and  x  +  y.     To  be  a  common  multiple  of  (1)  and  (2)  it  must 

contain  an  additional  factor  x  +  y  ;  that  is,  it  must  contain  {x  —  ?/),  {x  +  ?/), 

(x  +  y). 

To  be  a  common  multiple  of  (1),  (2),  and  (3)  it  must  contain  an  addi- 
tional factor  X  —y ;  that  is,  it  must  contain  the  factors  (x  —  y),  (x  -^  y), 
(x  +  y),  (x-y). 

The  product  (x  -  y)  (x  +  y)  (x  +  y)  (x  -  y)  ={x-  yY{x  +  yY  is  the 
lowest  common  multiple  of  (1),  (2),  and  (3). 

Rule  for  finding  Lowest  Common  Multiple.  To  obtain  the  lowest 
common  multiple  of  a  set  of  expressions : 

(1)  Find  the  j^rime  factors  of  each  expression. 

(2)  Use  all  factors  of  the  first  expression,  together  ivith  those  fac- 
tors of  the  next  expression  tvhich  are  not  in  the  first,  those  of  the 
third  which  are  in  neither  the  first  nor  in  the  second,  etc. 

It  is  evident  that  in  this  manner  we  obtain  a  product  which 
is  a  common  multiple  of  the  given  expressions,  but  such  that  if 
any  one  of  these  factors  is  omitted,  it  will  cease  to  be  a  multiple 
of  some  one  of  the  expressions ;  that  is,  it  will  no  longer  be  a 
common  multiple  of  them  all. 

Thus,  if  in  the  example  above  either  of  the  factors  x  —  ?/  is  omitted,  the 
product  will  no  longer  be  a  nmltiple  of  x-  —  2  xy  +  y-. 

ORAL   EXERCISES 

Find  the  L.  C.  M.  of  the  following  sets  of  expressions : 

1.  3,    5.  6.    ah,    he,    ac.  11.    2x-{-3,   x-i. 

2.  3,    4,    (•).  7.    3  a,    2  b,   4  c.  12.    y/i  +  3,    m  -  o. 

3.  ('>,    48,    24.  8.    a  4-1,  a-1.  13.    ax"^,    b%    aW. 

4.  a,    h.  9.    h  -h  2,    h  +  3.  14.    x  +  4,    x  —  1. 

5.  a,    hk.  10.    1-a,    1-2  a.        15.    xyz,  yzv,  2  vx 


COMMON   MULTIPLES  159 

WRITTEN   EXERCISES 

Find  the  L.  C.  M.  of  the  following  expressions : 

1.  2.3-4;  3.7.8;  23 .  3  .  4. 

2.  5  x^y^,   10  Oz-^?/,   25  xhj. 

3.  2  ah,   Q>a\   4  62c. 

4.  x^  —  ?/2^    a.'2  —  2  a;//  +  2/2. 

5.  x  —  y,    x  +  y,    xr  —  f-- 

6.  4-a;2,    2-aj,    2  4- x. 

7.  a2_^2a6  +  ^',    a^-2ah^h\ 

8.  .^2  +  3  X  +  2,    a.'2  -  4,  x"  - 1. 

9.  25a-2-l,    125.^3-1. 

10.  2.^2- 7. «  4- 6,   4  3;2_ij[^_^6^ 

11.  x^-y^,   x-y,    a;2  +  a-?/ +  2/2. 

12.  a^  —  y^,    x^  +  y^,    x^  —  y~. 

13.  5a;2  +  7.T-6,   a;2_;[5^,_34 

14.  .^•3  +  ^/'j    a'2  —  2/2,    (a;  —  yy. 

15.  3  a5c,   a2 ._  4  ac  +  4  c^,   a  —  2c. 

16.  .t2  —  1,    x  +  1,    a;2  4-80;  +  7. 

17.  4  a;32/ -  44  cc22/ +  120  a;^,   3  a3a;2  -  22  a^^  +  35  a'. 

18.  x'^ -\- 2  xy  +  7f,   2  ax"^  —  10  ax  +  12  a. 

19.  3  ?Ax2  -  21  6.1- +  36  ?>,    a;2- 5x4-4. 

20.  5  a-^s  _  5  a2c2,    h'' +  2bc  +  b +  c  +  c\ 

21.  15  ra.i*^  4- IG  c'-a.x- +  c-a,    2  caxi- +  10  cax -\- S  ca. 

22.  af*  — 6a;4-4,    a;  —  4,    a;2  — IG. 

Solution.     By  means  of  the  remainder  theorem,  we  find, 

a;3_Ca:  +  4  =  (a:-  2)(x2+  2  a: -2). 
Hence  the  L.  C.  M.  is 

(x  -  2)  (a-  4  4)  (.r  -A){x-  +  2x-  2). 

23.  .r^ -6.1-24-5,    ;r4-3,  a'2-1.' 

24.  .1-2  4- 5  .1- —  1,    .^2  — Ga-4-3,    x—1. 

25.  a;2  — 5x-4-6,    a;2  -  7  a;  +  12,    a;2— 9iy4-20. 


CHAPTER    X 
ALGEBRAIC  FRACTIONS 

125.  Fractions.  In  arithmetic  a  fraction  such  as  f  is  usually 
regarded  as  2  of  the  3  equal  parts  of  a  unit. 

However,  a  fraction  such  as  ^  cannot  be  regarded  in  this 
way,  since  a  unit  cannot  be  divided  into  3i  equal  parts.     ^ 

•  •  •  K 

indicates  that  5  is  to  be  divided  by  3^;  i.e.  —  =  5  -~  3i 

In  algebra  a  fraction  is  usually  regarded  as  an  indicated 
division  in  which  the  numerator  is  the  dividend  and  the  denom- 
inator is  the  divisor. 

Thus,  -  is  understood  to  mean  a  -^  b. 
b 

Terms  of  a  Fraction.  The  numerator  and  denominator  are  to- 
gether called  the  tei^ms  of  the  fi'action. 

Operations  on  algebraic  fractions  are  performed  in  accord- 
ance with  the  same  rules  as  in  arithmetic. 

17.      ^  „       1-4.      2      4         a     2  a    m     mx 
lor  example,  lust  as-  =  -,  so  -  =  — ,    — = — , 

^    '*'  3      6'        b      2b'    n      nx' 

and  a-hb_  (x-y)(a-^b) 

c-\-d      (x  —  y)(c-\-d) 

ORAL  EXERCISES 

Supply  the  missing  numerator  or  denominator  in  each  of  the 
following: 


1. 

2 

1 

2       4 

3 

4      12 

3. 
4. 

1 

a      ka 

«  _ 
b       cb 
160 

5. 
6. 

m 

n       3  n 
1 

a      —2  a 

ALGEBRAIC    FRACTIONS  161 


7. 

6      2 
9 

8. 

^7 

y 

g     aa;  +  a?/  _ 

10. 

15_ 

20      4* 

11. 

y 

12.     ^  +  2/^=     . 
a(x-{-7j)       a2 

1  ? 

x^  —  y"^    _ 

17. 

O.'^  —  ?/^ 

a{x  -  y)       a 

x^  —  y^     x-\-y 

14 

1 

18. 

1 

a  —  1      1  —  a 

6  +  2     (6+3)(6+2) 

1  fi 

1 

• 

19. 

a;2  _  ,y2   _ 

a  +  1      (a  +  l)(a 

-1) 

(ic  +  ?/)2      a;  +  2/ 

16. 

a;2  4-  3  a?  +  2      x 

+  2 

20. 

.'«^  —  ?/2 

x^  —  1  {x  —  yyx  —  y 

The  preceding  examples  illustrate 

Principle  XVII 

126 .  Rule .  Both  terms  of  a  fraction  may  he  multiplied 
or  divided  by  the  same  7vini%ber  without  changing  its  ualue^ 

HISTORICAL   NOTE 

Fractional  Notation.  Our  present  method  of  representing  fractions  is- 
the  result  of  a  long  historical  development.  The  Babylonians  used  OOths- 
only.     Thus  \  was  called  30  sixtieths,  J-  was  called  20  sixtieths,  etc. 

The  Egyptians  used  only  fractions  with  a  numerator  1.  The  denom- 
inator only  was  written,  a  dot  over  it  indicating  that  it  represented  a 
fraction.  Thus  3  would  mean  \.  The  use  of  "  unit ' '  fractions  only  neces- 
sitated the  reduction  of  other  fractions  to  this  form.  Thus  |  =  ^  +  yi^, 
and  f  =  1  +  ^V 

The  Greeks  used  one  accent  to  indicate  the  numerator  and  two  to  indicate 
the  denominator.     Thus  2'  3"  meant  |. 

The  Romans  used  a  duodecimal  system  of  fractions.  That  is,  all 
their  fractions  were  twelfths.  The  Hindus  wrote  f  in  the  form  |,  not 
using  the  bar.  Alkarismi  (see  page  101)  used  the  bar,  writing  the  fraction 
in  its  present  form.  Decimal  fractions  came  in  much  later  than  the 
common  fractions.  Simon  Stevinus  of  Bruges  in  Belgium  (1548-1620)  was- 
the  first  to  treat  decimal  fractions  systematically. 


Ib2  FRACTIONS 

REDUCTION   OF   FRACTIONS   TO  LOWEST   TERMS 

127.    By  Principle  XVII  any  factor  common  to  the  numerator 
and  denominator  of  ai fraction  may  be  cancelled.     That  is 

ak      a 
bk~  b 

.3 
Thus  2.g-4.5^2-4-  5.       2^-  3^-  :fx^  3a;  ^3 jr. 

3-7  .  11         7  .  11  '         2-i.3.;|2       2-4       8  ' 

2  4 

a;^  -  7  x  +  12  ^  (^^^)(a:-4)  _  a;  -  4 
ic2  -  5  X  +  6       (x  -  2)  (x^^)  ~  x  —  2 

Lowest  Terms.     If  the  terms  of  .a  fraction  have  no  common 
factor,  the  fraction  is  said  to  be  reduced  to  its  lowest  terms. 

ORAL  EXERCISES 

Reduce  the  following  fractions  to  lowest  terms : 

1-    *•  10. '^-y       ,.       lo      ^+-3        - 

14  x^  —  2xy-^y 


2 


2  1' 


6. 


6. 


2X-\-4: 

X 

x^ 

—  3x'^-{-x 

x-1 

{X 

-3)(^-l) 

x-^2 

x" 

+  3  a?  +  2 

x-2 

x^ 

-3x-{-2 

x-\-  1 

x^ 

+  3a;  +  2 

X 

-.'/. 

11.    ^-±^^ 20. 

x^  -{-2  xy  +  y"^ 

12.  i^±yl,         21. 

13.    ^i+-? 22. 

.^•2  +  5  rt'  -h  6 

14.    ^^ 23. 

a;^  —  4  a;  —  5 

15.  ^'-^ 24. 

0^2-7  X  -h  12 

16.    ^^—^ 25. 

x^  —  5  .X'  +  4 

17.    ^i±i 26. 


.^2  +  5  a;  +  4 

x  +  1 
x'-y^  -     a;2  +  4a;  +  3  '"    x''-Sx-\-15 


9.    ^ — ^.  18.    — ^~^^       .         27 


a.'2 

+  4a;  +  3 
a;-2 

a.'2 

-5a;  +  6 
x-{-2 

a;2 

-\-5x-\-6 
x-\-2 

a;2 

+  6a;+8 
x-2 

x^ 

~6x-\-S 
a;4-3 

x" 

+  5  a;  +  6 
a;-3 

X"' 

—  5x-^6 
a!  +  3 

x' 

+  8a;-f  15 
a;-3 

REDUCTION   TO    LOWEST   TERMS  163 

WRITTEN   EXERCISES 

Reduce  the  following  fractions  to  lowest  terms  : 

^      3  »  9^ .  2V  ^g     5  x^i/  -  12  x'y''  +  7  a?y'' 

*    2* .  53  .  9*'  '  6  xY  +  '6  xhf- 

3     ^V^.  18     ^  ^'  -  28  x^  +  48  j:^ 

xyh^'  '      2x'-Sx^-tiJx'' 

a^b^'  '              3ab'-3abc'' 

g     x""  +  2  a.7/  +  ?/2  2Q     7  x'/- 133  a;?/ +  126  a; 

x^  —  y^  '          15  xy^  —  36  xy  +  21  a; 

g     x^-\-7  x-  30  2^     20  .1-3  4- 20  .-«^;/ +  5  a;?/^ 

■    a^2_7^^_pj^2*  '          60x'-lox^y^ 

„  x?-f  „„       3a64-3o62c2 

7.    ^ •  22.    • 

2  a;2  -  3  a;?/  +  y""  27  a^ft^  _|_  27  a^jjc 

64-63  4a3_42a2  +  20a 


16  -  8  6  +  62  2  a'*^^  -  20  a^ft^ 

^  x^  +  21z'  24.    (:t--l)(x-2)Gr-3)(.r-4)^ 

x?/— 5x+3  2/2;— 152;  (.c— l)(a;— 3)(a;— 3)(aj— 4) 

10.  1-2160^  25.    (.i-^-^/^)(.-«^  +  2a.7/+y^). 

a;  —  4  2/  —  6  ex  +  24  c?/  (x^  —  2  x?/  +  2/^)('^'  +  y) 

^^     l^bz-2bx+ax-laz  ^^     (x'^- l)(x'^ +  1)(3  x^ +  3) 
x2- 49:^2             •        •  3(x^-l) 

^2         3a2-29a  +  56  ^^  (x2-4)(x2  +  4)(x  +  5) 

63-9a-7m  +  m(/        "    (x -2)(x2+ 3  x-10)Oi''- 16)' 

^3  a{x-yy  28     (2.r^  +  3x-2)(2x2  +  7x  +  6) 

(x2  -  ?/2)(x  -  2/)  *     (4  x2  +  4  X  -  3)(x2  +  4  X  +  4)  * 

14     __iiJl2^_.  29     (2x2-3x4-1X2  ^'^  +  3x4-1) 
■    4.-^2  4-24  x  + 36  ■  (x2-l)(4x2-l) 

^g     a2-3a-364-a?>  3^     (2x24-5x4-2)(3x24-10x  +  3)_ 
•    ■       (a2-62)(a-3)    *  *    (3  x2  + 7x +  2)(2x2  4- 7  x4- 3) 


164  FRACTIONS 

REDUCTION   OF   FRACTIONS   TO   A   COMMON   DENOMINATOR 

128.   Illustrative  Examples.     1.   Reduce  ^  and  ^  to  a  common 
•denominator. 

Solution.    I  =  i;     i  =  f-    Is   6    a   common  multiple   of    2   and    3  ? 
Is  it  the  least  common  multiple  of  2  and  3  ? 

2.  Reduce  -  and  -  to  a  common  denominator. 

a  b 

Solution.     -= —  ;    -=— . 
a    ab      h    ah 

How  is  the  common  denominator  related  to  the  denominators 

a  and  b  ? 

2  3  4 

3.  Reduce     -, r—. —, to    a   common 

denominator. 


a  +  1'     (a  +  l)(a-f2)'     a +  2 


Solution.    The  required  denominator  is  the   L.  C.  M.  of  the   given 
denominators  ;  that  is,  (a+l)(a  +  2). 

Hence,  _2_  =  _2(^±2L_  ^nd  -J-=  ,     *(\+'^       • 
a  +  1      (a  +  l)(o  +  2)  a +  2      (a  +  l)(o  +  2) 

already  has  the  required  denominator. 


(a+l)(a  +  2) 

ORAL  EXERCISES 

Reduce  each  of  the  following  to  a  common  denominator: 

a     b  ab     ac 

«     1     1     1  K      «      ^ 

Z.      — ,    — ,    — •  5.      — ,     — • 

a     b     c  be     ac 

„     1       1  ^      a      b 

b.     — ,    — • 

xy     xz 


a     —a 

10.        1     , 

a; -hi 

1 

x-^2 

■•  .43. 

1 
a  +  4 

10              « 

c 

b     c  —  d 


7. 

1       1 

1 

a6'    be 

ae 

a      b 

c 

8. 

be    ae 

ab 

9. 

m      n 

s 

Qcy     xz 

yz 

13. 

1 
1-A:' 

1 
k-1 

14. 

1 

x-{-2' 

1 

x-2 

15. 

a 

b 

tt-f  6 

a-b' 

REDUCTION  TO   A   COMMON   DENOMINATOR  165 

129.    By  the  formula  "=— ,  Principle  XVII,  both  terms  of 

b      bk 

a  fraction  may  be  multiplied  by  the  same  number. 

In  this  manner  any  fraction  may  be  changed  into  an  equal 
fraction  whose  denominator  is  any  given  multiple  of  the  de- 
nominator of  the  given  fraction. 

'^'  4     4.5'     «  +  &      (a+b){a  +  h)      {a  +  b)^' 

Any  two  or  more  fractions  may  therefore  be  changed  into 
equal  fractions  which  shall  have  a  common  denominator,  namely, 
a  common  multiple  of  the  denominators  of  the  given  fractions. 

Illustrative  Example.      Reduce 

(1)  ^;    (2)   ^;  and  (3)  ^|i+^ 
x-\- 1  x  —  1  ar  — 1 

to  fractions  having  a  common  denominator. 

The  L.  C.  M.  of  the  denominators  is  (x  —  l)(x+  !)•  Multiply  the 
numerator  and  denominator  of  each  fraction  by  an  expression  which  will 
make  the  denominator  of  each  new  fraction  (x  —  l)(x  +  1). 

Thus  ^^  =  (x-l)(x-l)  ^    x'~-2x+-i     .  ,^. 

x+1      (x  +  l)(x-l)      (x  +  l)(x-l)' 

x  +  l  ^(x  +  l)(x+l)  ^        (x+1)-  (2) 

x-1      (a;-l)(a:  +  l)      (x  +  l)(x- 1) ' 

2x  +  3  ^         2x  +  3  ^gv 

x2-l       (x  +  l)(x-  1)' 

It  is  best  to  indicate  the  multiplication  in  the  common  denominator, 
since  this  makes  it  more  easily  apparent  by  what  expression  the  terms  of 
a  fraction  must  be  multiplied  in  order  to  reduce  it  to  a  fraction  with  the 
required  denominator. 

ORAL  EXERCISES 

Supply  the  missing  numerators  : 

1  ^ 

1.    -  =  — -'  3. 


X     x\x-l)  x-l-5      x2-f7x-f-10 

2.     — ^:— = 4.     — ^^  = 

X  +  1      of  -\-S  x-\-7  X  —  o      .y2  —  2  a;  —  15 


166  FRACTIONS 

130.  The  Three  Signs  of  a  Fraction.  There  are  three  signs  in 
connection  with  a  fraction :  (1)  the  sign  of  the  fraction  itself, 
(2)  the  sign  of  the  numerator,  and  (3)  the  sign  of  the  denom- 
inator. Any  two  of  these  signs  may  be  changed  simultane- 
ously without  changing  the  value  of  the  fraction. 

Thus,     (1)|  =  5--.      (2)|=-f^.      (3)^--^- 

If  either  the  numerator  or  the  denominator  is  a  polynomial,  its 
sign  can  be  changed  only  by  changing  the  sign  of  each  term  in  it. 

En     ^  +  ^  —  ~'^~^—       (i  +  l'> 
c  —  d      —  c-\-  d  — c  +  d" 

If  either  the  numerator  or  the  denominator  is  in  the  factored 
form,  its  sign  can  be  changed  only  by  changing  the  sign  of  an 
odd  number  of  its  factors. 

Ea    (a^  +  y) (y -  y)  ^  (a;  +  y) (y  -  x) 
(a-\-b){a-b)      {a -{- b){b  -  a)' 

This  is  useful  in  examples  like  the  following  : 

CC  ~\~  i-  X  1 

Example.     Keduce  — — — ,    ,    and  to  fractions 

1  —  X     x"^  —  1  X  -f-  1 

having  a  common  denominator. 

x  +  l  _-x—l  _  (x+l)(—x-l)  _  — a;2-  2x  -  1 
l-x~   a:-l    ~   (x-\-l)(x-l)   ~         x'^-l  ' 

a;2_i     x^-1  x  +  l      {x+l)(x-l)      x^-1 

ORAL  EXERCISES 

Apply  the  sign  changes  shown  in  (1),  (2),  and  (3)  above  to 
each  of  the  followiuGf  fractions: 


o 


(■4)    (a-b)(c-d)  ^^^ -G 

—  X  {d  —  a)(b—a) 

(5)    ^ (8)  -(«-&)(ft-c), 

(1  -  x)  (X  +  2)  (c  -  d){d  -  a) 

(d)        ^>-«  (<))       {-a)(-h)(-c)^ 

c  +  d                       ^           d-c  ^         d(-e)(-f) 


(1) 

b 

a 

—  c 

(2) 

a 

-  b 

c 

-d 

(3) 

-  a 
1   ,j 

REDUCTION  TO   A    COMMON   DENOMINATOR  167 

WRITTEN  EXERCISES 

Reduce  each  of  the  following  sets  of  fractions  to  equivalent 
fractions  having  a  common  denominator : 

_a;  +  3  4  ^     a  -\-b       a  —  b  a 


y  9  r.  .         o  "•       1  .'/...      r\9' 


5. 


X  —  y     x^  —  2  xy  -\-y^  h  —  a     (a  +  &)^     ct^  —  ^^ 

a-1  a  +  1  .  1  1 


'S  —  X  x-\-4:  „         a  +  1  a  —  1 


.x.2_9.y_|_20'  7x2-26a;-8  cC^-2ah^-W    a^^2ah+h^ 

2        a;-l     ;r  +  l  ^1  1  1 

o. 


?>  —  x     oj+l'   it- —  3  a?—W    h  —  a     cr  +  ab-^-lr 

a  be 


5a^-4.a-12'    a'' -\- 4.  a  -  12'    a-2 
mr  d        , ,  i?r  1  . «       a  6 


10.   -^^^^,   -^ —     11.   7— — ^ r,  -^ 12 , 


7?i  — 1     l+^i  {Ii-\-r)(m—l)    R-\-r             n  —  a    n  —  b 

13     W(T-Q)      V  ^g    _R^          1              1 

wiQ-t)  '    w  '  a-A'    R-\-A'    A-a 

,^          V              V  1             ,a     ^      ''         1             i?.f 


V—  V     V-\-  V     V^  —  v^  X      y    x  —  y     x-{-y 

131.  Reducing  Integral  Expressions  to  the  Fractional  Form. 
Since  any  number  may  be  written  as  a  fraction  with  the  de- 
nominator 1,  the  above  process  may  be  used  to  reduce  an 
integral  expression  to  the  form  of  a  fraction  having  any  desired 
denominator. 

Thus,  3  =  §-1^;   x-y  =  (^-y)(^'^-l),  etc. 

5  a;-  —  1 

R    rf>  1  X  II 

Illustrative  Example.     Reduce  5  a;,  — '- ,    and  ^^ •-    to 

x^  —  1  x—1 

fractions  having  a  common  denominator.     The  lowest  common 

denominator  is  x"^  —  1. 

Thus,  5,.^5a^(a:-3-l)^5a-3-5a;       5x-^l  ^  5 x -  1    ^^^ 

x2  -  1  a-2  -  1  x^^l       x^-\ 

2  X  —  If  _  {2  X  —  y)(x  +  I)  _  2x^  +  2x  —  xy  —  y 
x-1   ~    {x-  l)(x+l)  ~  x:^-X 


168  FRACTIONS 

WRITTEN  EXERCISES 

Eeduce  the  following  expressions  to  fractions  having  a  com- 
mon denominator : 


x^ -\- 2  xy -\- y"^  x—y    x-\-y 


2.  l^—^^  2h-c^  2c^-2.       7.  3a-26-c,    — 5— ,  ^— • 

x—y       x-\-y  a  —  h    b  —  c 

3.  l+a  +  a2,    ^L±i.  8.  .t*  - 1,    x^-1,    ^^^. 

a  — 1  ic—  1 

4.  x''-\-xy  +  y'^,    ^^±1.  9.  a;2  +  2a;?/+2/^  ^^,  — ^< 

a;  —  ?/  '  a; +  2/     it'-?/ 

5.  x'-xy^y^   x^-y\  -^.    10.  x^y,  x-y,    ^=^    ^^+i. 

ic+2/  x^-\-y^    x—y 

11.    _:^,    ,.  12.    -^,    E-r.  13.    T,    ^±i. 

a-A'  a-A'  '        2 


ADDITION   AND   SUBTRACTION  OF   FRACTIONS 

132.   Rule  for  Adding  or  Subtracting  Fractions.    By  Principle 
III,  §  16,  we  know  that 

— ■ — =--!-- and = . 

C  C  C  C  G         C 

Heading  this  in  the  reverse  order  we  have  the  formula  for  add- 
ing and  subtracting  fractions : 

a  ,  b      a  +b       ^  a     b      a  —  b 

-  H-  -  =  — ■ —  and = • 

c     c         c  c     c         c 

From  this  formula  we  get  the  following  rule. 

(1)  Reduce  the  fractions  to  a  comDvoii  dniominator. 

(2)  To  add  the  fractions,  add  tJve  numerators  and  place 
the  sum  over  the  common  denominator. 

(3)  To  subtract  the  fractions,  subtract  the  numerators 
and  place  tlw  remainder  over  the  common  denominator. 

(4)  Reduce  the  resulting  fraction  in  each  case  to  its 
lowest  terms. 


ADDITION   AND   SUBTRACTION  169 


1  2 

Example.     Add and 


a  -1-1  a-\-S 

.Solution.  -±-  = ^±i ;  -^  =       ^^^  +  ^)      ■ 

a  +  l      (a+l)(a  +  3)'  a+3      (a+l)(a  +  3) 

Hence        I      ,      ^     ^         ci  +  S  2(a  +  1) 

'   a  +  1      a  +  3      (a+l)(a  +  3)  (a+l)(a+3) 

^g +  3 +  2(0  +  1)^         3a  +  5 
(a +  !)(«  + 3)       Ca  +  l)((z  +  3)' 


ORAL  EXERCISES 

Perform  the  following  additions  : 

1.    l  +  i.  5.    ?+     1 


a      b  a      —  a 


a6      ac  1  —  ^'      k  —  1 

3.    -i^-f-i^-  T.    ^+      1 


a  +  1      a  — 1  n  —  4      n  — 1 


a.--|-la;-|-2  a;-|-2a;  —  2 

3  2 

Example.     From  subtract 

X  —  4:  a;  -f-  3 

3  2     ^       3(a;  +  3) 2(x  -  4) 

x-4     x+3      (x-4)(x  +  3)      (x-4)(x  +  3) 

^  3(x  +  3)-2(x-4)  x  +  17 

(x-4)(x-h3)  (x-4)(x  +  3) 


ORAL  EXERCISES 

Perform  the  following  subtractions  : 

1.  i-i.  4.    1 


a6  a;-f-lic-|-2 

11  2       1 

2.    — -— .  5.    -^i---. 

ab      ac  —  a     a 

3     _1 3_.  6     _1 1_ 

'a-l-la  —  1  *1  —  A:A:— 1 


170  FRACTIONS 

Example,     Add     ^^^^nd  ^' +  ^  ^^  +  ^' . 
a  +  6         a2  -  2  a6  +  62 

Solution.    The  L.  C.  M.  of  the  denominators  is 

{a-b){a-h){a  +  h). 

a-h  ^  {a-h){a-  b)(a-  b)  ^  qs  -  3  a'^b  +  3  ab^  -  fcs 
a-\-b      {a  +  b){a-b){a-b)      (a  +  b)(a  -  b)(a  —  b)' 
and     «^  +  2  a?>  +  &^  ^  (a  +  &)  (a'^  +  2ab  +  b^)  ^  ft-^  +  3  g^ft  .^^  3  ^52  ^  53  ^ 
a2_2a6  +  &=^       (a  +  b){a  -  b)(a  -  b)       (a -\- b)(a- b)(a  -  b)' 

Adding  the  numerators,  we  have  2a^  +  6  ab'^ ;  whence  the  sum  of  the 
fractions  is  2a^  +  Qab^  ^ 

(a +  &)(«-  b)(a-b)' 

WRITTEN  EXERCISES 

Perform  the  following  additions  and  subtractions : 

1.   -^ —.  n.   _^  +  -J_  +  l. 


1 

1 

x-2 
1 

x  +  1 

2 

a  +  4 
1 

a  —  S 
1 

1  —  a  '  a  —  1 

2.  -^^-^^-  12.     -^ -  +  - 

x  —  yyx 

3.  -    ^----^ 13.     .^_  +  f^_i, 

ic  — 11  —  a;  x  —  y      y      y 

4.  ^-i.  14.  !+«-*. 

ic      .7  X      y      z 

5.  ' — •  15.     -H \--' 

a—b      a+h  x      y      z 

6.  -^ ^.  16.  '' 


a 


k-1      k-\-l  (a;-l)(a;  +  3)      x-{-3 

7     _4 3  17.       1      ■      1      •  1 


c-2      (c-l)(c-2)  x+y    x-y     (x-{-y)(x-y) 

8.     1-1  +  i.  18.       1      ■      1  1 


a      6      c  a  +  1     a  — 1     (aH-l)(a  — 1) 

9.    i  +  l_i.  19.    3  +  2       a  +  6 


a6      6c      abc  4      7      3  a  —  6 

10.     -^ ^      +i-  20.      ^-?/    _  ^  +  y 

a  -h  6      a  —  6      a  (a?  +  2/)^      7?  —  y"^ 


ADDITION   AND   SUBTRACTION  171 


x'^  —  9x-\-lS.x  a-\-l  a  —  1 


x'^  —  13x-\-36     4:  —  x  a2  +  a  +  l      a^  — a  +  l 


3      a  +  6a-6  1-^1+6      1-62 

23     _A_4._5 ?_  29     >-g  +  l  ■  a^  +  1  ■  3a:  +  2 

23. 32  "^22.34      2^.33  ■    ic-2      icH-2       a;2-4 

04          a2— 952           a^—6ab  x  —  1  _  x -{- 1      o^^-_5 

a2+6a6+9  62~  a2-9  62'  '    a;  +  l~a;-la;2-l* 

25.    a;  +  y  .  x-y      ^  ^^        ?/'       ,      V y     , 


x  —  y     x-{-y  2/^  —  1      2/  +  1      1  —  2/ 

26.    ^^- J L.  32.    i-1-      1^-1. 


ic2_2/2      cc  —  2/      ic  +  T/  aJ      2/      x  —  y      x  +  y 

33     1         ^  2a2         — g 

2      6-a     62_^2"^5  +  ^* 

a;2  +  4>Ty  1     x 


aj^  +  2/'        X  +  y      x^  —  xy  -^  y"^ 

35  ct  —  3  a  —  1 a 

a2-3a  +  2     a2-5a  +  6a2-4a  +  3 


36. 
37. 
38. 
39. 

40. 


X  ,       2  x 


a;2 -5aj- 14      x-7      ic2-9a;  +  14 

a b 

ac-^ad—bc  —  bd     a^  —  2ab  -\-  b^ 
1  1.4 


2/2 +  82/ +  16     2/(2/4-4)      2/X2/  +  4) 
a4-2       ,         a  — 4  a4-2 


a2-a-6      a^-l  a  +  12      d'-2a-S 

? ^+^- 


41.    1 +  ^ 


42. 


(a;-l)(a;  +  2)      (ic  +  2)(x-3)      (a;-l)(3-ic) 

I I + I 

(a  _  h){b  -  c)      (6  -  a)(c  -  d)      (b  -  c){c  -  d) 

4  a-1        .a2-lla-3 


43. 

a-3      a2  +  3a  +  9  a^-27 


172  FRACTIONS 

MULTIPLICATION   OF   FRACTIONS 
133.    Fractions  are  multiplied  as  in  arithmetic. 

rru  2^5      2.5      10 

Thus  -  X  -  : 


Similarly, 


3     7     3.7     21 


a      c  _a  •  c  _ac 
b     d~  b  '  d~~ bd 


That  is,  we  have  the  following 

Rule.  TJw  product  of  two  algebraic  fractions  is  a  frac- 
tion whose  numerator  is  tlie  product  of  the  given  numer- 
ators and  whose  denominator  is  the  product  of  the  given 
denominators. 

134.  Steps  in  Multiplying  Fractions.  The  steps  in  multiply- 
ing one  fractional  expression  by  another  are : 

(1)  Reduce  mixed  expressions  to  fractions. 

(2)  Factor  each  nmiierator  and  each  denominator. 

(3)  Cancel  all  factors  common  to  numerators  and  denomi- 
nators. 

(4)  Midtijyly  the  remaining  factors  in  the  numerators  for  the 
numerator  of  the  product,  and.  those  of  the  denominators  for  the 
denominator  of  the  product. 

These  steps  are  illustrated  in  the  following  examples. 

(1)  5x8=^  =  ?;       (2)  l|x2|  =  |xf  =  ^=4. 

3 

(3)         /i+«U-i^  =  ^«x-^  =  ^^i^::^=6. 
^  ^  \    ^b)      h-\-a         h         h  +  a     ^CM-O 

(^\  4x  +  12  3.r-3  ^(g^^^)  •  SO^^TT) 

^^  x-i+x-2     4x2+4x-24      (a^^-'r)(a:  +  2).<(^K3)(x-2) 

3  ^3 

~(x  +  2)(x-2)      ^2-4" 


MULTIPLICATION  173 

ORAL  EXERCISES 

Perform  the  following  multiplications  : 

^     ab      c  „    abc   ,  hx  _    rnxv      ny 

1.    — X--  3.    — X 5.   ^X— ^• 

c       a  xy       ac  nyc      mx 

2  ex      10  a  hi/     dx  xy      r^s 

7.    (a  4-  o)  X  — 


a}  —  h"^  (a^^\^{a  —  b)      a  —  b 

«/         7\.       1  -.rt'^  +  V       X  —  y 

8.    (a— 6)X— -•  12.    —^-^  X  — — ^ 

a^  —  b^  X—  y      x^  -{- y^ 

■9.  ^iL^x^^!:^'.  13.  ("1+-^    " 


a  +  6a  —  6  \        xj     x  -\-l 

10.    (a-l)X— ^-  14.  -^         X  6(yi  + 1). 


a^  -  1  3(?i  +  1) 

1  2 

^  +  /  ^^     (a-2)(a+3) 


11.    (a;  +  2/)x--^-  15. ^-^  x  3(a-2). 


WRITTEN  EXERCISES 


Find  the  following  indicated  products  and  reduce  the  frac- 
tions to  simplest  forms : 


x-y 

x-\-  a 


1.    (i_a)xl+i^-^.  2.    (a:3_y)x'^±l. 

a  —  1  a?  —  ?/ 

3.    (ic2  — 2a;a  +  a2)  x 

X—  a 

*•    -/Vx7X(^-^-llr.  +  18). 

X^  —  DX-\-  O 

6.    0T2+9a;H-18)x         ^~^ 


x''-2x-W 
7.    (l-a.'^)x-^-=^-  8.   (27a'-l)x        ^^^ 


l  +  x  +  x"  "  -^     9a24-3a  +  l 


174  FRACTIONS 


Perform  the  following  multiplications  : 
9.    (a2  +  a&4-62)x4^-     11.    4^  X    ""' ~  ^ 


10.  (1  -  «  +  ce)  X  ^.      12.  4+1 X  «;±|- 

^  a^  +  1  a^  —  6^      a-  —  b^ 

13  3c  ^^Jc-3)(c  +  3) 

(3-c)(3+c)  6ccl 

3a-\-b      (3x  +  2)(2a-b)  ^^     a-b       a'-¥ 

'   2  +  3aj       (a-{-b){3a-hb)  '  a' -  b'     a'-b^ 

(a-^2b)(a-{-2b)      (2a-b){a-2b) 
a -2b  (3a +  ?>)(« +2  6) 

3^Y      6a^  jg    5  a{a  -b)       9(a  +  6)'^ 

2  2/^2      9a.-3*  '   3c(a  +  6)      lo(a2-62) 

^g       12c^6         35(c'^4-c6  +  6'). 
5(c3_63)  14r62 

2^    ?/2  +  3y  +  2^^y2-7y  +  12 

„,     x'-x     2a;2  +  4a;  +  2  _„     a^- 10a +  16      a  +  3 

21.    X ' ' — •  22. -M ' . 

x^-1         3a;2+6aj  o}  +  Q>a  +  \)        a^  -  4 

x^  -\-xy  —  xz      (x  —  zy  —  y-     xy  —  y'^-\-yz 

24  3(0^  +  4)^        ^         {x-iy 

4(a;  +  4)(a;-7)      3{x  +  4.){x-l) 

2g     o?-\-b  «^  -  36  a      (a-16)(a-3) 
a2-7a-144  ^  a(a- 4)(a  +  2) ' 

2g    3  a{a  +  7)(a  -  5)      6(a  +  3)(a  +  10) 
7  6(a  +  3)(a  +  7)      (/(a  -  r))(a  -  10) 

3^2_2^_1     2«2^5^-3     4^2^10^  +  4 
2«2_^^-l       3^2  _^  7^  +  2       Art''-2t-2 

^^      6a;2-7a;  +  2       (Sx^-^x-l      10iB2  +  3a;-l 
10x2-7a:  +  l       6^2+ re -1        5a^-4a;-l 


DIVISION  175 


DIVISION   OF   FRACTIONS 

135.   Fractions  are  divided  by  the  same  rule  as  in  arithmetic. 

3 
rpr,  282^92-^3 

4 

CI-      1     1     •      1     u        a     c      a      d     ad 

Similarly  in  algebra,   -^-  =  --x     =-— . 

b      d      0      c      be 

That  is,  we  have  the  following 

Rule.     To  divide  hy  a  fraction,  invert  the  divisor  and 
multiply  hy  the  fraction  thus  obtained. 

Steps  in  Dividing  Fractions.     The  steps  in  dividing  by  a  frac- 
tion are : 

(1)  Reduce  all  mixed  expressions  to  the  fracMoiKd  form. 

(2)  Factor  each  numercdor  and  each  denominator. 

(3)  Invert  the  fraction  hy  which  you  are  dividing. 

(4)  Now  proceed  as  in  multiplying  fractions. 

These  steps  are  illustrated  in  the  following  examples. 

/,>.  ah h a^ 3>^^  _     a 

^  x^-Sx  +  2  '  x-l~  (x^^X^-^^)         ^     ~x-2' 

^2y        ax -ay   ..        y^^(i(j>^l)  y^      ^     =        ^ 

b(x  +  y)  '  '        b{x  +  y)      x^     b{x  +  y) 


(3) 


fl+^U(«^-&2)^^:^x -^ = ^~ 

\        aj  a         {a^^^){a-h)      a(a  —  b) 


In  Example  (3),  the  divisor  is  an  integral  expression,  a^—  b^. 

q2  ^2  .  ,  ][ 

This  may  be  written ,  and  this  inverted  is 


a2-62 


ORAL  EXERCISES 

Perform  the  following  divisions: 


a 

a2 

b 

b^ 

1     ^^ 

X 

y' 

y 

'    f~^f'  '    h^  '   ¥ 

^^^.  6     —  — — 


176  FRACTIONS 

WRITTEN   EXERCISES 

Perform  the  following  divisions  : 

uo  .  abc  ,   be  -     nxy  .   ny 

7712/      mx 

24  a6  .  16  a& 
25  c    ■     15    ' 
4A:2  2A: 


5 

a&  .  bx 

c       c 

abc 
x-1  ' 

ab. 

^y    . 

a  —  b 

xy. 

a  +  l 

3(g 

^+1) 

a-2  ' 

a 

+  4 

10. 

11. 

12. 


—  9  aftc'^  a6 

aW  ab 

(x-{-yy  '  x-^y 


Find  the  following  indicated  quotients  and  reduce  the  frac- 
tions to  their  lowest  terms  : 


13.    ^-y'-^(x''  +  xy-\-y'').  14.    t±Jl  ^(x^ -y^). 

x  +  y  x  —  y 

.'u^  —  2  ic  +  1 

1  —  9a;^ 

17.  — — '-(x^  —  4:X  —  o). 

18.  x'-Wx-^-  39  ^.  2  _  ^,  _  155X 

a;2  _  8  a;  +  15       ^  ^ 

19.  ^"  +  ^"-^^-^^^-H-(cfl?-3c-d.r  +  3(?). 
x?/  —  4  a;  —  3  ?/  +  12 

20      ^^  +  "'^  +  ^'■'^  +  «&  ^(,^2  +  ax  -  5 a-  -  5 a). 
aj2  +  6a;  —  3  a;  —  3  6 

21.  • ■ 'r-{or  —  xr -^  6s  —  xs). 

mx  —  m  —  nx  -f  n 

22.  ^^l^l^(x'  +  9x  +  8). 
x^  —  \)  X  —  2Z 


MULTIPLICATION  AND   DIVISION  177 

EXERCISES   IN  MULTIPLICATION  AND   DIVISION 

Perform  the  following  indicated  opprations : 

1  «'  +  ^'    V  "  +  ^ ^  3     a;2-6>T-16  .  a;^+9a;  +  14 

'    a2-9  62       a  +  5*  '    x''+Ax-21  '  x'-Sx +  15' 

2  x''-\-x-2  .      a;'^  +  2a;2        ^  a;^  - 1  a;2-25 

ic2_3a,     •  ^2^9^_36*      •    a;2_4iC-5  ^a;2  +  2a;-3* 

a;^  4-  9  a^  +  18  y^  .  a;^  4-  6  a;?/  +  9  y^ 
a;^  —  9  xy  +  20  ^/^  xy"^  —  1 2/^ 

g       3  g^  -  9  g^  -  54  g'^     .  g^  +  8  a^  +  15  ft  . 
9g3-117g2  +  378a  '  3g2-33g  +  84* 

g^  -  11  g  4-  30      g^  -  3  g  .         g"  —  9 


6. 


11. 


12. 


a3_6g2  +  9g      g2-25      g2  +  2g-15 

_     aj2-10a;  +  21  .  a;2-8a;H-15 
aj'^  +  a;  —  56        a;^  +  4  a;  —  32 

g     g2-62  b{a-b)       .  g""-2g&  +  ^'. 

gft^a?        g2  +  2g6  +  52  *       b(a-\-b) 

10     ft  +  &        ft" -6^     .  {a-by(a-^by 
ab        3(g2  +  62)  *      3ahj-\-8ay' 


5x'-5x^  x^-dx^  +  Sx'' 


7x''-56x-63      14a;2  +  14aj-1260 

8?A.V  +  4)0y  +  5)  .    y(y  +  4)(7/  +  8) 
2X2/  +  5)(2/-7)     •  23(2/-7)(^  +  ll) 


^3     a^(a.'  -  2)2      (x  +  2)(x  -  3)  .  a;^(a^  -  3)(a;  -  5) 
(a; +  2)2       (a^-2)(a;-7)  *    (a;-5)(a;-7) 

j4     a'b%c  +  5)(c  -  4)      (c  -  8)(c  +  9)  .  g6(c  +  9)(c  - 1) 
(c_4)(c-8)         (c  +  4)(c  +  7)  •     (c  +  7)(c  +  l) 

15     21a;2-f-23a;-20      6ar^-lla;-10  .  7a;2  +  17x-12 
10a;2-27a;  +  5   ^    3a;2  +  2a;-5     '     5a;2  +  9a;-2 

4  52-17  6  +  4      10  y-  _  21  6  +  9      3  ?)2  -  5  ?>  +  2 
*     662-76+2  ^562-236  +  12      462-56  +  1* 


178  FRACTIONS 

COMPLEX  FRACTIONS* 

136.   A  complex  fraction  is  one  which  contains  one  or  more 
fractions  in  its  numerator  or  denominator  or  in  both. 


„  a         ,    X  -\-  1      X—  1 

E.g.  --J    and        ^  ,     • 

a  X  —  I     X  -\-\ 

Simplifying  a  Complex  Fraction. 

A  complex  fraction  is  said  to  be  ^nmplified  when  it  is  reduced 
to  an  equal  fraction,  in  its  lowest  terms,  whose  numerator  and 
denominator  are  in  the  integral  form.  The  following  examples 
show  how  such  reductions  are  made : 

14-1     «  +  l 

a         a     _q  +  la  —  l_a+l         fi     _a+l. 


1- 


1~^_1  a  a  ^         a  —  1      a  — \ 


a         a 
This  result  may  also  be  obtained  directly  by  multiplying  both  terms  of 
the  given  fraction  by  a  which  is  the  L.  C.  D.  of  all  the  small  fractions  in 
the  complex  fraction.      This  gives 

1+1      «(l  +  l)      a  +  ? 

,       1~/,       l\~        a~  a  —  1 
1 ail 1      a 


a        \        aj  a 

1  1  X-  1  +  x-{-l 


2  x-\-l      x-1  _  x^-1  _    2x     .       2      ^^ 

_1 1_  ~  x+l-(x-  1)      x'^  -1  '  x'^-1 

X  —  1     X  -\-  1  x"^— 1 

By  multiplying  the  terms  of  the  given  fraction  by  (x  +  l)(a;  —  1),  we 
may  also  get  directly 

(x^^)(x-l)      {x+\){x^^^) 

>H^  >-^l  _x-  1  4-g  +  1  -^^-x 


(x+ l)r;t>--T)      {y^^y^){x-\)     x  +  l-a:+l       2 
After  a  little  practice,  this  cancellation  can  be  done  mentally , 

X \  A-  X  -\-\ 

thus  writinG: — — ^^—  at  once. 

^  a;  +  l-a;+l 

*§§  136,  137  may  be  omitted  without  destroying  the  continuity. 


COMPLEX   FRACTIONS  179 

137.    Rules  for  Reducing  Complex  Fractions. 

Rule  1.  Reduce  both  numerator  and  denominator  to 
a  single  fraction  by  ])erform1n^  the  operations  indicated. 

Then  divide  the  numerator  by  the  denominator. 

Rule  2.  Multiply  both  numerator  ojid  denominator 
by  the  lowest  common  denominator  of  all  the  small 
fractions. 

Then  reduce  the  resulting  fraction  to  its  simplest  form. 

Eule  2  is  applied  in  each  of  the  following : 

1  »+;-l_3+2-l_4^2 

f  +  l-i     4  +  5-3     6     3' 
Here  we  multiply  both  numerator  and  denominator  of  the  given  fraction 
by  6,  the  L.  C.  M.  of  the  small  denominators,  2,  3,  and  6. 

14-1-4--1-       24-1-1-4       7 
2.   Similarly,  K±i±|  =  jL±L±l  =  7 

Here  we  multiply  numerator  and  denominator  by  8,  the  L.  C.  M.  of  2,  4, 
and  8. 

WRITTEN   EXERCISES 

Eeduce  each  of  the  following  complex  fractions  to  its 
simplest  form : 

1. 


2. 


3. 


4. 


\  +  x 

1+i 

X 

1-^ 

b 
1-f-a* 

+  1 

5. 

6. 

7. 
8. 

4  +  «t' 

9. 

10. 

11. 

H     hd 
c      cD 

1  +  t 
c 

X 

4      «-^ 

2 

x^—  if 

4 
x-\-y 

a^-Sb^ 

27 

X 

"  +  2 

x-S 

x-^2 

x  +  S 

x-2 

X 
X 

2 

m  -\-n 
3 

x-S 
x-S 

X  +  4: 

x-\-2  ' 

x-\-S 

a -2  b 

M 

1  +  6? 

X  —  4: 

111  —  n 

-1 

x-\-2 

D^  X-4: 


180  FRACTIONS 


Simplify  the  following 


1  +  1 


12. 


X—  4      x 


15. 


X 


x^-lx  +  12 

1  16. 


13.    ^^      1 


1+x 


a 


14.        ,      a 


17. 


a;-l 

x-2 

3 
ic  —  3 

2 
a.' -3 

4     • 
x-1 

2a;2  4-2 

a;-2a^ 

1 

1 

a;-3 

a  —  b 

l-2a;2 
b 
"^a  +  6 

a 

a 

a +  35  a  +  6a  —  6 

REVIEW  QUESTIONS 

1.  How  is  a  fraction  defined  in  algebra? 

2.  In  what  ways  may  a  fraction  be  changed  in  form  with- 
out changing  its  valne  ?     State  Principle  XVII. 

3.  How  is  a  fraction  reduced  to  lowest  terms  ? 

4.  How  are  fractions  reduced  to  a  common  denominator? 

5.  What  are  the  three  signs  of  a  fraction  ?  How  may 
these  be  changed  without  affecting  the  value  of  the  fraction  ? 
Explain 

8_        8_      — 8_— 8      A^^_    ^   _      —  a _  —a 
4"~      ^~     ~^~Zri^^    l~~'Z~j)~        5~ ~  -b' 

6.  State  the  following  principles  in  words  : 

/1\  o_f^o  fiys   a     b_a±b 

v-l;  7  —  — 7  ^-^j  ~±-  — 

0      mo  c      c         c 

7*  Explain  how  a  complex  fraction  nuiy  be  reduced  to  a  sim- 
ple fraction  in  two  different  ways.    Which  is  the  shorter  method? 


CHAPTER   XI 
EQUATIONS   INVOLVING  FRACTIONS 

138.    Example  1.     Solve  n +-+-  =  88.  (1) 

Solution.  Multiplying  both  members  of  equation  (1)  by  6,  which  is 
a  common  multiple  of  2  and  3,  these  denominators  are  cancelled,  and 
we  get  at  once 

Qn  +  ^n-\-2n  =  528.  (2) 

Uniting  terms,  11  n  =  528.  (3) 

Dividing  both  sides  by  11,  n  =  48.  (4) 

The  object  is  to  multiply  both  members  of  the  equation  by  a 
number  that  ivill  cancel  each  denominator.  Hence  the  multiplier 
must  contain  each  denominator  as  a  factor. 

Evidently  12  or  18  might  have  been  chosen  for  this  purpose,  but  not 
8  or  10.  6  is  the  smallest  number  which  will  cancel  both  2  and  3,  because 
6  is  the  L.  C.  M.  of  2  and  3. 

Example  2.     Solve  -  +  _1_  =  ^—1 —  .  (1) 

X      X  —  1       x{x  —  1) 

Solution.  Multiplying  both  members  of  equation  (1)  by  the  L.  C.  D., 
x{x  —  1),  we  have. 

Cancelling,  we  have 

x-l  +  2x=l.  (3) 

Hence  3  x  =  2  and  aj  =  |. 

Here,  as  in  Example  1,  the  members  of  the  given  equation  are 
multiplied  by  an  expression  which  cancels  each  denominator. 

181 


182  EQUATIONS   INVOLVING   FRACTIONS 

139.    Clearing  of  Fractions.    The  process  explained  in  the  fore. 
going  solution  is  called  clearing  of  fractions. 
As  another  illustration  solve  the  equation 

-A ^  = 4 

a;  4-1      x-1      {x  +  l){x-l)  ^  ^ 

Here  the  lowest  common  multiple  is  {x  +  \)(x—  \). 

When  we  multiply  by  (ic  +  l)(x  —  1),  the  denominator  x  +  1  is 

X  +  1 


cancelled,  thus, 


x^\ 


Similarly,  -  (£L±il(^::ll)  =  _  (a:,  +  i), 


and  ii^)^^^4. 

Hence,  equation  (1)  becomes  4(x  —  1)  — (cc  +  1)=  4.  (2) 

By  i^,  4  a;  -  4  -  ic  -  1  =  4.  G'>) 

By  F,  and  ^15,  8 a:  =  9  (4) 

ByD|3,  a:  =  3.  (5) 

Check.     Substituting  x  =  3  in  (1)  we  get 

1  -  1  =  I  or  i-  =  i 
After  a  little  practice  the  cancelling  of  the  denominators  can 
be  dona  mentally,  as  we  multiply  by  the  least  common  denomi- 
nator in  clearing   of   fractions.       For  example,   in  this  case, 
equation  (2)  may  be  written  at  once. 

The  foregoing  examples  illustrate  the  following 
Rule.     To  dear  an  equation  of  fractions,  muHiphj  hotli 
inemhers  by  the  L.  C.  D.  of  all  the  fractions  involved,  and 
then  cancel  the  denoininators. 

WRITTEN  EXERCISES 

Solve  the  following  equations,  indicating  the  principles  used 
at  each  step  in  Examples  1  to  10  : 

1.    -  4- -  =  5.  3.    — \- \n = h -o. 

2      3  4  32 

234  2^3      410 


FRACTIONAL   EQUATIONS  183 


5.    7a;  +  if +  ^+23  =  -  +  — +  5a;  +  113. 
7        5  5      5 


6.    4  n  H = ~ h  46. 

.7  2 


7    12  I  4(9a;  +  6)      2(3  +  11  a;)  ^5(4a:  +  4)      ^^^ 
"^3  5  3 

7  5 

g    5a  +  7      2aH-4  _  3a  +  9      ^ 
2  3  4 

10     1^  ^  —  5     10  ft  +  2  _  5  g  +  7      ^ 
3  4        ~       2  '^* 

11.    ^^±^  +  ^12^.iM  =  2a;  +  24. 


12.  ?-!  =  !.  15.    1+     '  ^ 


X      2      ic  a?      a;  —  1      re  (.t  —  1) 

111  4-^-2 

13.  —  +  — =  i.  16. 


3  a;      2  a;      6  x-{-lx-lx'--l 

14.  i+l=A+l.  n.   ^+    1  ^ 


4  ic      3x     6x      2  X  —  1      X  +  1      x"^  —  1 

18.  The  sum  of  two  numbers  is  12,  and  the  first  number  is 
I  as  great  as  the  second.     What  are  the  numbers  ? 

19.  The  smaller  of  two  numbers  is  -|  of  the  larger.     If  their 
sum  is  66,  what  are  the  numbers  ? 

20.  Find  two  consecutive  integers  such  that  j-  of  the  first 
minus  y^-  of  the  second  equals  9. 

21.  Find  three  consecutive  integers  such  that  ^  of  the  sum 
of  the  first  and  second  minus  ^  the  third  equals  5. 

22.  Find  three  consecutive  integers  such  that  |-  of  the  first 
plus  I  of  the  second  minus  -^^  of  the  third  equals  28. 


184  EQUATIONS   INVOLVING  FRACTIONS 

140.  Special  Cases.  I.  Sometimes  it  is  best  to  add  fractions 
before  multiplying  by  the  L.  0.  D.  as  in  the  following  example : 

a  1                        1111  ,,, 

Solve -  = -.  (1) 

Adding  fractions  on  the  right  and  left, 

a;-l-(x-2)^a;-3-(a;-4) 

(x-2)(x-l)      (x-4)(a;-3)' 
Simplifying  the  numerators, 

I = I (2) 

{x-2){x-\)      (x-4)(a:-3)  ^ 

Multiplying  by  the  L.  C.  D.  of  all  the  fractions, 

(x-4)(x-3)  =  (a;-2)(x-l).  (3) 

Hence,  solving,  x  =  2^.  (4) 

Check  by  substituting  x  =  2|  in  equation  (1). 

Note.  —  If  we  attempt  to  solve  this  equation  by  first  clearing 

of  fractions,  we  have 

(x-l)(x-4)(x-3)-  (x-2)(x-i)(x-S) 
=  (x-2)(x-  l)(x-3)-(x-2)(x  -  l)(x-4). 

The  solution  of  this  equation  is  much  more  laborious  than 
the  solution  of  equation  (2)  above. 

II.  Sometimes  it  is  best  to  multiply  by  the  L.  C.  D.  of  j^ci'^t 
of  the  denominators  first,  and,  after  simplifying,  multiply  by 
the  L.  C.  D.  of  the  remaining  denominators. 

c.  1  4f-3     t-2     2t-2  ,., 

Solve  — — : —  = •  (1) 

16  4         5t  +  2  ^  ^ 

Multiplying  by  16,  4 «  -  3  -  4(«  -  2)  =^^ — —•  (2) 

bt  +  2 

Hence,  5=-^^ — ~  '     •  (3) 

Multiplying  by  6  «  + 2,  25  f  +  10  =  32  t  -  32.  (4) 

Hence,  solving,  t  =  6.  (5) 

Check  by  substituting  ^  =  G  in  equation  (1). 

In  this  example,  try  to  solve  by  clearing  of  fractions  com- 
pletely at  the  outset  to  see  whether  the  plan  of  solving  here 
given  is  simpler. 


FRACTIONAL  EQUATIONS  185 


2 


10. 
11, 


WRITTEN  EXERCISES 


1. —  = ■ h4.        Ans.  x  —  \^. 

5  15         6x-S  ^ 


7a;  +  l      14a;-22^11a;  +  5 
12  24  8a.'- 28' 


2  8  3a;  +  2 

^     7^  +  3      21^  +  9      17^-3  ,  o        A        ,  iQ 

^  -         — +2.      ^TIS.  i  =  — i|. 


5  15  3^  +  11 

11  ?;  - 15      33  ?;  +  15  ^  5  ?;  +  5 
10  30  v-5 


Ans.  V  =  4. 


6.    -  +  - = -.  Ans.  x  =  Si. 

x  —  2      o  —  x     x  —  4:      x  —  o 

12  12, 

1. = Ans.  x  =  ^. 

x-1      2a;-fl     x-2     2x-l  ^ 

-    x  —  2     ic  —  3     x  —  4:.x—6 
cc  —  3      a;  —  4      a;  —  5      6  —  x 

9  9  5  5 


x  —  7  x-2      .T— 8      x  +  1 

x  +  11  2(a;  +  6)  ^      x-1 

.T  +  5  .T  4-  3            ic  +  3 

3a;-4  4.T-1           a.-^  +  44       ^^ 

a.' +  5  a; +  4       x'^9x  +  20~ 


PROBLEMS   LEADING   TO   FRACTIONAL  EQUATIONS 

1.  What   number    must   be    subtracted  from  each    term  of 
the  fraction  ii  so  that  the  result  shall  be  equal  to  i? 

2.  What   number   must  be    subtracted   from   each  term  of 
the  fraction  |-i-  so  that  the  result  shall  be  equal  to  |? 

3.  What  number  must  be  added  to  each  term  of  the  frac- 
tion ^  to  obtain  a  fraction  equal  to  J4  ? 

4.  What  number  must  be  added  to  each  of  the  terms  in  the 
fractions  -J  and  ^  in  order  to  make  the  resulting  fractions  equal  ? 


186  EQUATIONS   INVOLVING   FRACTIONS 

5.  There  are  three  numbers  such  that  the  second  is  4  more 
than  9  times  the  first,  and  the  third  is  2  more  than  6  times 
the  first.  If  ^  of  the  third  is  subtracted  from  ^  of  the  second, 
the  remainder  is  3.     Find  the  numbers. 

6.  There  are  three  numbers  such  that  the  second  is  2  more 
than  9  times  the  first  and  the  third  is  5  more  than  11  times 
the  first.  The  remainder  when  -i-  of  the  third  is  subtracted 
from  i  of  the  second  is  one.     Find  the  numbers. 

7.  What  number  must  be  subtracted  from  both  the  nu- 
merator and  the  denominator  of  the  fraction  f  in  order  to 
make  the  result  equal  to  |-  ? 

8.  What  number  must  be  subtracted  from  both  numerator 
and  denominator  of  the  fraction  |-  in  order  that  the  fraction 
may  be  increased  threefold  ?     Ans.  2i. 

9.  What  number  added  to  both  numerator  and  denomina- 
tor of  the  fraction  f  will  double  the  fraction  ? 

10.  Find  a  number  of  two  digits  in  which  the  tens'  digit  is 
3  greater  than  the  units'  digit,  and  such  that  if  the  number  is 
divided  by  the  sum  of  the  digits,  the  quotient  is  7. 

11.  In  a  number  of  two  digits  the  units'  digit  exceeds  the 
tens'  digit  by  4,  and  when  the  number  is  divided  by  the  sum  of 
its  digits  the  quotient  is  4.     Find  the  number. 

12.  Illustrative  Problem.  B  can  do  a  piece  of  work  in  8  days, 
and  A  can  do  it  in  10  days.  In  how  many  days  can  they 
do  it  working  together  ? 

Since  B  can  do  the  work  in  8  days,  in  1  day  he  can  do  \  of  it, 
and  since  A  can  do  it  in  10  days,  in  1  day  he  can  do  ^^^  of  it.  If  x 
is  the  number  of  days  required  when  both  work  together,  then  in  1  day 

they  can  do  —  of  it.     Hence  we  have  the  equation, 

X 

8       10      X 


REVIEW   QUESTIONS  187 

13.  B  can  do  a  piece  of  work  in  12  days  and  A  can  do  it  in 
9  days.     How  long  will  it  take  both  working  together  to  do  it  ? 

14.  A  pipe  can  fill  a  cistern  in  11  hours  and  another  in  13 
hours.     How  long  will  it  require  both  pipes  to  fill  it? 

Arts.  o||  hours. 

15.  B  can  do  a  piece  of  work  in  a  days  and  A  can  do  it  in  b 
days.     How  long  will  it  take  both  together  to  do  it  ? 

16.  A  cistern  can  be  filled  by  one  pipe  in  20  minutes  and 
by  another  in  30  minutes.  How  long  will  it  take  to  fill  the 
cistern  when  both  are  running  together  ? 

17.  A  pipe  can  fill  a  cistern  in  12  hours,  another  in  10  hours, 
and  a  third  can  empty  it  in  8  hours.  How  long  will  it  require 
to  fill  the  cistern  when  they  are  all  running  ? 

18.  A  man  can  do  a  piece  of  work  in  18  days,  another  in  21 
days,  a  third  in  24  days,  and  a  fourth  in  10  days.  How  long 
will  it  require  them  when  all  are  working  together  ? 

Ans.  4g^  days. 

REVIEW  QUESTIONS 

1.  What  is  meant  by  clearing  an  equation  of  fractions  ? 

2.  What  principle  is  used  in  clearing  an  equation  of  frac- 
tions ? 

3.  By  what  expression  must  both  members  of  an  equation  be 
multiplied  in  order  to  clear  the  equation  of  fractions  ? 

4.  Is  it  always  best  to  multiply  at  once  by  the  least  common 
multiple  of  all  the  denominators  ? 

5.  In  the  first  example  solved  on  page  184,  state  what  ad- 
vantage was  gained  by  adding  the  fractions  on  each  side  be- 
fore clearing  of  fractions. 

6.  In  the  second  example  solved  on  page  184,  what  advantage 
was  gained  by  clearing  of  fractions  partially  at  first  ? 


CHAPTER   XII 
RATIO  AND  PROPORTION 

141.  Ratio.  A  fraction  is  often  called  a  ratio.  Thus  -  may 
be  read  the  ratio  of  a  to  b,  and  it  may  also  be  written  a  :  b. 

Terms  of  a  Ratio.  The  numerator  is  called  the  antecedent  of 
the  ratio,  and  the  denominator  the  consequent.  The  antecedent 
and  consequent  are  called  the  terms  of  the  ratio. 

Proportion.  An  equation,  each  of  whose  members  is  a  ratio, 
is  called  a  proportion. 

Thus,  -  =  -  is  a  proportion.     A  proportion  is  usually  written 
b      d 

a  :  b  =  c  :  d,    or    a  :  b  :  :  c:  d. 

It  is  read  the  ratio  of  ato  b  equals  the  ratio  of  c  to  d,  or  briefly, 
a  is  to  b  as  c  is  to  d. 

Means  and  Extremes.     The  four  numbers  a,  b,  c,  and  d,  when 

written  -  =-,ov  a:b  ::  c:d,  are  said  to  be  in  j^roportion.     Then 
b     d 

the  two  end  terms,  a  and  d,  are  called  the  extremes  of  the  pro- 
portion, and  the  two  middle  terms,  b  and  c,  the  means. 

HISTORICAL  NOTE 

Ratio  and  Proportion.  Tlie  subject  of  ratio  and  proportion  was  studied 
fully  by  P^uclid  (300  b.c.)  in  connection  with  geometric  magnitudes,  but 
of  course  all  his  results  apply  equally  well  to  numbers.  Euclid,  however, 
did  not  regard  a  ratio  as  a  fraction  and  his  treatment  is  exceedingly  com- 
plicated as  compared  with  the  modern  treatment  in  which  a  ratio  is 
recognized  as  a  fraction. 

In  1631  Oughtred,  an  Englishman,  wrote  the  proportion  a  :  h  =  c  :  (f  in 
the  form  a  :b  :  :c  :d.  Before  his  time  this  proportion  was  written 
a—  b  —  c  —  d.  John  Wallis  (see  opposite  page)  brought  the  symbol  :  : 
into  common  use  and  it  has  surviveil  to  the  present  time,  though  its  mean- 
ing is  exactly  the  same  as  that  of  = . 

188 


John  Wallis  (1616-1703)  was  an  English  clergyman  who 
made  contributions  to  mathematics,  logic,  and  grammar.  He 
was  educated  at  Emmanuel  College,  Cambridge,  and  was  after- 
ward chosen  fellow  of  Queen's  College, 

During  the  period  of  the  conflict  between  King  Charles  I  and 
Parliament,  Wallis  was  an  adherent  of  the  party  of  the  latter, 
and  displayed  surprising  talent  in  deciphering  intercepted  papers 
and  letters  of  the  Royalists. 

Wallis  was  the  author  of  numerous  works  relating  to  logic. 
English  grammar,  and  especially  to  mathematics.  His  work  on 
algebra,   De  Algebra  Tractatus,  contains  a  history  of  the  subject. 


IMPORTANT  PROPERTIES   OF   A   PROPORTION         189 


IMPORTANT   PROPERTIES   OF   A  PROPORTION 

142.    Transforming  a  Proportion.     Starting  each  time  with  the 
proportion,  a:b  : :  c:  d,  we  deduce  the  following  results  : 

Case  I :  ad  =  be. 

ft  C 

Proof.     Writing  «  :  6  :  :  c  :  d,   in   the  fractional  form,  -  =    ,  we  clear 

h     d 

of  fractions  and  obtain  ad  =  be. 

That  is  :  If  four  niunbers  are  in  proportion,  the  product  of  the 
means  equals  the  product  of  the  extremes. 

Case  II :  b  '.  a  :.  d  .  c. 

We  are  to  show  that  if  -  =  -,  then-  =  - . 

h      d  a      c 

Proof.    From  Case  I,  we  have  he  =  ad. 

Dividing  both  sides  by  ac^  we  get  -  =  -  ,  or  6  :  a  ::  cZ  :  c. 

a      c 

This  process  is  called  taking  the  proportion  by  inversion. 
Case  III :  a  :  c  : :  b  :  d. 

We  are  to  show  that  if  -  =  - ,  then  -  =  - . 

b      d  c      d 

Proof.     From  Case  I,  we  have  ad  =  be. 

Dividing  both  sides  by  cd,  we  get  ^^-  =-,  or  a  :c  :  -.b  :  d. 

c      d 

This  process  is  called  taking  the  proportion  by  alternation. 
Case  IV:  a  +  b  :  b  \  :  c  +  d  :  d. 

We  are  to  show  that  if  -  =  -,  then  ^^^^  =  ^^t_^. 

b     d'  b  d 

Proof.     We  have  -  =  - . 
b      d 

Adding  1  to  both  sides,  we  s^et  -+  i  =-+  i. 

b  d 

Hence  9l±1  =  1±A^  ot  a  +  b  :  b  ::  c  +  d  :  d. 
b  d 

This  process  is   called  taking  the    proportion  by  addition, 
or  by  composition,  as  it  is  sometimes  called. 


190  RATIO    AND   PROPORTION 

Case  V:  a  —  b  :  b  :  :  c  —  d :  d. 

We  are  to  show  tliat  if  -  =  - ,  then  = 

b      d  b  d 

Proof.     We  have  -  =  -. 
b     d 

(1  c 

Subtracting  1  from  each  side,  we  get 1  = 1. 

h  d 

Hence  ^^^^  =  ^^,  or  a- b  :  b  : :  c  -  d  :  d. 
b  d 

This  process  is  called  taking  the  proportion  by  subtraction, 

or  by  division,  as  it  is  sometimes  called. 

Case  VI:  a  -\-  b  :  a  —  b  : :  c  -\-  d  :  c  —  d. 

AVe  are  to  show  that  if  -  =  -,  then  — ^t_^  =  -^ — 

b      d  a  —  b     c  —  d 

Proof.     We  have  from  Cases  IV  and  V, 

(1)     ^±^=^-+^;and   (2)   «^^=:^^^l^. 
b  d  b  d 

Dividing  equation  (1)  by  (2), 


Jt}         a  —  b        ^        c  —  d 

Hence  ^-±-^  =  ^^tl ,  or  a  +  b  :  a  -  b  ::  c  +  d  :  c  -  d. 
a  —  b      c—  d 

This  process  is  called  taking  the  proportion  by  addition  and 
subtraction,  or  by  composition  and  division. 

CaseVII.    If  ^  =  ^  =  ?,then^+^  =  ^, 
b      d     f  b  +  d-\-f     b 

Proof.     Let  -  =  -=-  =  k;  then  a  =  bk,  c  =  dk,  e  =fk. 
b     d      f 

Hence  a  -{-  c -\-  e  =  bk  +  dk  -{-fk  ={b  +  d  +  f)k, 

and  «  +  ^  +  ^  =  /c-=.^  =  "^g. 

&  +  (/+/  b      d     f 

That  is,  The  aum  of  the  antecedents  is  to  the  sum  of  the  con- 
sequents as  any  antecedent  is  to  its  consequent. 

Mean  Proportional.     If  a  :  ^ :  :  ^  :  a;,  then  b  is  called  a  mean 
irroportional  between  a  and  x. 

Fourth  Proportional.     If  a  :  b::c:x,  then  x  is  called  di  fourth 
proportional  to  a,  b,  and  c. 


FURTHER  PROPERTIES  OF  A  PROPORTION     191 


WRITTEN  EXERCISES 

1.  If  ad  =  he,  show  that      =  — .         Hint.     Divide  by  hd. 

h       d 

2.  If  ad=hc,  show  that-=-.    3.  If  ad=bc,  show  that  -  =  -. 

c      d  c     a 

4.  If  ad  =  be,  show  tha,t  -  =  -. 

b      a 

5.  If  ?  =  «,  show  that  ^^+i?  =  5+i?. 

b      d  a  e 

^     TO  a      e      -,         iiiCfc— 6      e  —  d 

6.  If  -  =  --,  show  that  — —  = . 

b      d  a  e 

7.  If  «  =  £ ,  show  that  "^^  =  '-^: 

b      d  a-{-b      c  -\-d 

8.  If  «  =  i,  show  that  ^L±i?  =  cizz*. 

b      d  c  -\-  d      e  —  d 

9.  It    -  =  -,  show  that  — - —  =  -. 

b      d  e  -\-d      e 

10.  If -  =  -,  show  that  ^^^t^  =  -. 

b      d  b-^d      b 

11.  If  -  =  -,  show  that  =  -• 

b      d  e—  d      c 

12.  If  -  =  -,  show  that  ^^:^  =  -. 

b      d  b-d      b 

13.  Solve  the  equation  — =-  for  each  letter  in  terms  of  all 

^  b      d 

the  others.  If  a  =  3,  b  =  o,  c  =  8,  find  d.  If  b  =  7,  c  =  9, 
d  =  3,  find  a.  If  e  =  13,  d  =  2,a  =  5,  find  b.  If  d  =  50,  a  =  3, 
6  =  -7,  find  c. 

14.  Find  a  fourth  proportional  to  3,  5,  and  7 ;   also  to  9,  5, 
and  1 ;  and  to  3,  —  2,  and  —  5. 

15.  Is  4  a  mean  proportional  between  2  and  8  ?     Why  ? 
Is  16  a  mean  proportional  between  4  and  32  ?     Why  ? 


192  RATIO   AND   PROPORTION 

PROBLEMS   nrVOLVmG   RATIOS 

1.  Which  is  the  greater  ratio,  -^ or  ^-^ ? 

Hint.  Reduce  the  fractions  to  a  common  denominator  and  compare 
numerators,     {d  is  a  positive  number.) 

2.  Which  is  the  greater  ratio,  ^L±l^  or  A±^l  ? 

^  a4-86        a  +  106 

3.  Which  is  the  greater  ratio,  -  or  ,  if  b  and  c  are  posi- 

h       b  -\-  c 

tive,  and  a  less  than  6  ?  a  equal  to  6  ?  a  greater  than  b  ? 

4.  Find  two  numbers  in  the  ratio  of  3  to  5  whose  sum 
is  160. 

Hint.     Call  the  numbers  x  and  160  —  x. 

5.  Find  two  numbers  in  the  ratio  of  2  to  7  whose  sum  is 
-108. 

6.  Find  two  numbers  in  the  ratio  of  3  to  —  4  whose  sum 
is  - 15. 

7.  What  number  added  to  each  of  the  terms  of  the  ratio 
-f  makes  it  equal  to  4|  ? 

8.  What  number  must  be  added  to  each  term  of  the  ratio 
Jy  to  make  it  equal  to  the  ratio  f  ? 

9.  What  number  added  to  each  of  the  numbers  3,  5,  7,  10, 
will  make  the  sums  in  proportion,  when  taken  in  the  given 
order  ? 

10.  Two  numbers  are  in  the  ratio  of  2  to  3,  and  their  sum  is 
GO.     Find  the  numbers. 

11.  What  number  must  be  subtracted  from  each  of  the 
numbers  7,  8,  9,  and  12,  so  that  the  resulting  differences  shall 
form  a  proportion  when  taken  in  the  given  order  ? 


SIMILAR   TRIANGLES  193 

SIMILAR  TRIANGLES 

143.  Similar  Triangles.  Triangles  are  called  similar  if 
they  have  the  same  shape. 

Thus  the  triangles  ABC  and  DBF 
of  the  figure  are  similar.  Note  that 
AB  and  DB  have  been  divided  into  7 
and  3  equal  parts,  respectively.  Hence 
the  ratio  of  these  sides  is  ^. 

What  is  the  ratio  of  the  sides  BC  and  B  D  A 

BE?  Of  the  sides  CA  and  ED?  This  is  stated  as  follows: 
Tlie  lengths  of  the  pairs  of  corresponding  sides  of  two  similar  tri- 
angles form  a  p)roportion. 

m.  ^  •                          -^     ^B      CB       CA 
That  IS,  we  may  write = =  — —  • 

'  ^  DB     EB     ED 

Note  that  AB,  BC,  •••  represent  the  lengths  of  these  sides. 

WRITTEN  EXERCISES 

1.  If  in  two  similar  triangles  the  sides  of  the  first  are  11, 
13,  and  16,  and  the  side  of  the  second  which  corresponds  to  11 
in  the  first  is  33,  find  the  other  sides  of  the  second  triangle. 

Solution.    Let  x  represent  the  length  of  the  side  corresponding  to  the 

one  whose  length  is  13.     Then  —  =  — ,  and  11  a;  =  13  •  33,  or  z  =  39. 

13      11' 

In  this  manner  find  the  remaining  side. 

2.  If  the  sides  of  a  triangle  are  4,  6,  and  10,  and  one  side 
of  a  similar  triangle  is  9,  find  the  remaining  sides  of  the  second 
triangle,  if  the  given  side  corresponds  to  the  side  4. 

3.  Solve  Example  2  if  the  given  side  in  the  second  triangle 
corresponds  to  6. 

4.  Solve  Example  2  if  the  given  side  of  the  second  triangle 
corresponds  to  10. 


194  RATIO   AND   PROPORTION 

5.  A  triangular  field,  one  of  whose  sides  is  20  rods,  has  an 
area  of  80  square  rods.      Find  the  area  of  a  similar  field  whose 

corresponding  side  is  45  rods. 

It  is  found  in    geometry    that    if   a  line 

divides  one  angle  of  a  triangle  into  two  equal 

angles,    it  divides  the  opposite  side  into  two 

jy         parts  which  are  in  the  same  ratio  as  the  other 

two  adjacent  sides  of  the  triangle. 

That  IS,  m  the  figure  777:  =  -^^^^- 
B" "A  DC      BG 

6.  Knowing  this  fact,  how  many  of  the  lines  AD,  DC,  BC, 
and  AB  must  you  measure  in  order  to  find  the  rest  of  them  ? 

7.  If  in  the  figure  AD  =  6,  DC=  9,  and  BC=  12,  find  AB. 

8.  If  in  the  figure  BC  =  18,  AB  =  12,  and  AD  =  6,  find  DC. 

This  fact  about  geometry  enables  us  in  some  cases  to  find 
the  distance  between  two  points  without  measuring  it  directly. 

9.  If  in  the  figure  CD  divides  the  angle  at  G  into  two  equal 
parts  and  if  you  know  the  lengths  of  ^ 

AD,  DB,  and  BC,  show  fully  how  to 

find   the   length   of    a   straight   line 

across  the  pond  from  O  to  ^  without  '  ^ 

measuring  it  directly. 

10.  If  jB(7  =  75  rods,  ^Z>  =  50  rods, 
and  AD  =  60  rods,  compute  AC.  ^  h^^-^-^=r^  A 

11.  If  the  sides  of  a  triangle  are 
a,  h,  c,  and  the  corresponding  sides  of  a  similar  triangle  are 

a,  b,  c',  show  that  — — ^ — -! — ,  =  -:• 

a'  +  &'  +  c'      a' 

12.  Two  corresponding  sides  of  two  similar  triangles  are  in 
the  ratio  13  :  14.  Show  that  the  perimeters  (sum  of  the  sides) 
of  the  triangles  are  in  the  same  ratio. 


SIMILAR  TRIANGLES 


195 


13.  The  perimeters  of  two  similar  triangles  are  in  the  ratio 
32 :  35.  Two  sides  of  the  first  triangle  are  8  and  12.  Find 
two  sides  of  the  second  triangle  corresponding  to  the  given 
sides  of  the  first. 

14.  In  the  figure  each  of  the  triangles  I  and  II  is  similar 
to  the  original  triangle.     From  these 
triangles  read  three  proportions,  using 
the  principle  of  §  143. 

15.  It  follows  that  the  triangles  I 
and  II  are  similar  to  each  other.     Head  three  proportions  from 

this  fact. 

16.  If  in  a  circle  two  intersecting  chords  are 
drawn,  as  in  the  figure,  it  is  known  that  ah  =  ccl. 
Form  a  proportion  from  this  equation.  If  a  =  9, 
h  =  S,  d  =  6j  find  c. 


REVIEW  QUESTIONS 

1.  Define  ratio,  proportion,  means,  extremes. 

2.  State  in  words  the  way  in  which  the  following  propor- 


tions are  derived  from  -  =  -• 

h      d 


a      c 


c      a 


(3) 


a+b     c+d 


(4) 


a  —  b      c  —  d 


(5)  ''  +  l>_c  +  d 


b  d  ^  '    a  —  b      c  —  d 

3.  Define  mean  proportional,  fourth  proportional. 

4.  What   can  you  say  of  the  corresponding  sides   of  two 
similar  triangles  ? 

Draw  a  large  figure  like  that  in  Example  14,  above.     Meas- 
ure the  sides  and  verify  the  proportions. 


CHAPTER   XIII 
LITERAL  EQUATIONS  AND   THEIR  USES 

144.  Advantage  in  using  Literal  Equations.  Some  of  the  ad- 
vantages of  algebra  over  arithmetic  in  solving  problems  have 
been  pointed  oat  in  the  preceding  chapters. 

A  further  advantage  is  set  forth  in  the  present  chapter ; 
namely,  the  opportunity  offered  in  algebra  to  summarize  the 
solution  of  a  whole  class  of  jyrohlems  by  solving  what  is  called  a 
literal  equation,  thus  obtaining  a  formula  which  may  be  used  in 
solving  all  particular  problems  of  this  type. 

For  example,  in  arithmetic  we  solved  many  problems  obtain- 
ing the  interest  when  the  principal,  rate,  and  time  were  given. 
We  now  see  that  all  of  these  can  be  summarized  in  the  one 
literal  equation,  which  is  o\w  first  formula  for  interest  problems : 

i  =  prt  (1) 

Furthermore,  the  rules  for  obtaining  the  principal,  the  rate,  and 
the  time  may  now  be  derived  directly  from  this  equation  by 
Principle  VI,  thus  obtaining : 

i       .,       ^        *        * 
rt  pt  pr 

Translate  each  of  these  formulas  into  a  rule  of  arithmetic. 

145.  Solving  a  Literal  Equation.  The  process  of  deriv- 
ing p  =  —  from  i  =^prt  is  called  solving  the  equation  i  =  prt  for 
})  in  terms  of  i,  r,  and  t,  or  simply  solving  the  equation  for  p. 
Similarly,  the  process  of  deriving  r  =  —  from  /  =prt  is  called 

soloing  the  equation  for  i;    and  deriving  t  =  —  is  called  solving 

pr 
the  equation  for  t. 

196 


INTEREST  PROBLEMS  197 

In  arithmetic  a  problem  is  said  to  be  solved  when  a  numerical 
answer  is  obtained  which  satisfies  the  conditions  given.  The 
solutions  thus  far  found  in  algebra  have,  for  the  most  part,  been 
of  this  sort. 

It  is  customary,  however,  to  say  that  a  j)roblem  has  been 
solved  in  the  algebraic  sense  when  a  formula  is  found  which 
gives  complete  directions  for  deriving  the  numerical  answer. 

Thus,  p  =  —  is  a  solution  for  the  principal  since  it  states  precisely  how 
tr 

to  find  the  principal  in  terms  of  interest,  rate,  and  time. 

It  is  thus  seen  that  from  the  literal  equation  .  =  jyrt  we 
obtain  the  complete  solution  of  every  problem  which  calls  for 
any  one  of  these  four  numbers  in  terms  of  the  other  three. 

In  modern  times  machines  are  used  extensively  for  compu- 
tation. The  algebraic  solution  of  a  literal  equation  gets  the 
problem  ready  for  the  computing  machine;  that  is,  it  gets  the 
formula  which  the  computer  must  use. 

I.     INTEREST  PROBLEMS 

Oral 

1.  If  $200  is  invested  at  5%,  what  is  the  amount  at  the 

end  of  one  year  ? 

This  problem  calls  for  the  amount^  which  is  the  sum  of  the  principal  and 
interest. 

2.  If  $500  is  invested  at  6  %,  what  is  the  amount  at  the 
end  of  one  year  ? 

3.  If  $1000  is  invested  at  4  %,  what  is  the  amount  at  the 
end  of  one  year  ? 

4.  If  $500  is  invested  at  5%,  what  is  the  amount  at  the 
end  of  2  years  ? 

5.  If  $1000  is  invested  at  6  %,  what  is  the  amount  at  the 
end  of  5  years  ? 

6.  If  i  =  prt,  how  would  you  represent  the  amount  in  terms 
of  p,  r,  and  r? 


198  LITERAL   EQUATIONS   AND   THEIR   USES 

Another  Interest  Formula.     If  a  =  amount,  then 

a=p+prt,=p(l+rt).  (2) 

This  is  our  second  formula  for  interest  problems. 
This  equation  may  now  be  solved  for  any  one  of  the  four 
letters. 

Thus,  solving  for  »,  we  have  p  =  — - — 

Using  this   equation  we  may  find  the   principal   when  the 
amount,  rate,  and  time  are  given. 

CI 

Translated  into  a  rule,  the  formula  p  =  - — —  reads  : 

1  +  rt 

To  find  the  principal^  divide  the  amount  by  1  plus  the  product 
of  the  rate  and  the  time. 

Written  Problems 

1.  Find  the  principal  if  the  rate  is  5  %,  the  time  2  years, 

and  the  amount  $  880. 

Solution.  _     a      _    880     _  880  _  g^^ 

^^  ~  1  +  r«  ~  1  +  .10  ~  1.1  ~        ' 

2.  Find  the  principal  if  the  rate  is  6  %,  the  time  4  years, 
and  the  amount  $  1488. 

3.  Solve   the    equation    a=p-\-2wt   for   t.     Translate    the 
resulting  formula  into  words. 

4.  Find    the  time  if  the  principal  is   $2500,  the  amount 
$  3100,  and  the  rate  4  %. 

5.  Find  the  time   if   the  principal   is  $5000,  the   amount 
S6050,  and  the  rate  6  %. 

6.  Solve   the  equation   a  =  p -\- prt   for  r  and  translate  the 
resulting  formula  into  words. 

7.  Find   the   rate   if   the   principal  is    $1800,  the    amount 
$  2124,  and  the  time  4  years. 

8.  Find  the   rate    if   the   principal   is   $3500,  the  amount 
$4340,  and  the  time  8  years. 


PROBLEMS    INVOLVING   MOTION         •  199 

"We  have  thus  used  the  formula  a  =  p  +  prt  to  solve  all  possible  types  of 
problems  where  the  amount,  principal,  rate,  and  time  are  involved.     The 

points  to  be  noticed  are  •  (1)  the  great  ease  with  which  the  equation  may 
be  solved  for  any  one  of  its  letters,  thus  obtaining  new  rules  of  aritlimetic  ; 
(2)  the  convenience  of  solving  problems  by  direct  substitution  in  a 
formula.     See  the  solution  of  Problem  1. 

II.     PROBLEMS  INVOLVING  MOTION 

146.  Formulas  for  Motion  Problems.  The  space  passed  over 
by  a  moving  body  is  called  the  distance,  and  the  number  of  units 
of  distance  traversed  is  represented  by  d.  The  rate  of  uniform 
motion,  that  is,  the  number  of  units  of  space  traversed  in  each 
unit  of  time,  is  called  the  speed  or  rate,  and  is  represented  by  r. 
The  number  of  units  of  time  occupied  is  represented  by  t. 

For  example,  if  a  train  runs  40  miles  per  hour,  in  5  hours  it 
will  run  5  X  40  miles. 

Again,  if  sound  travels  1080  feet  per  second,  in  5  seconds  it 
will  travel  5  x  1080  feet. 

In  each  of  these  examples  the  distance  passed  over  is  found 
by  multiplying  the  rate  by  the  time.  Using  the  symbols  d,  r, 
and  t,  we  have  the  first  formula  for  motion  : 

d=rt  (1) 

1.  Solve  the  equation  d  =  rt  for  t  in  terms  of  d  and  r,  and  for 
r  in  terms  of  d  and  t.     See  §  145. 

Translate  eaeh  of  these  formulas  into  words. 

It  is  to  be  understood  in  all  problems  here  considered  that  the  speed 
remains  the  same  throughout  the  period  of  motion  ;  e.g.  sound  travels  just 
as  far  in  any  one  second  as  in  any  other  second  of  its  passage. 

2.  If  sound  travels  1080  feet  per  second,  how  far  does  it 
travel  in  6  seconds  ? 

3.  If  a  transcontinental  train  has  an  average  of  35  miles  per 
hour,  how  far  does  it  travel  in  2\  days  ?  Here  we  have  given 
r  =  35,  t  =  2^  X  24,  and  we  are  to  find  d. 


200 


LITERAL   EQUATIONS   AND   THEIR   USES 


4.  A  hound  runs  23  yards  per  second  and  a  hare  21  yards  per 
second.  If  the  hound  starts  79  yards  behind  the  hare,  how 
long  will  it  require  to  overtake  the  hare  ? 

If  t  is  the  number  of  seconds  required,  then  by  formula  (1)  during  this 
time  the  hound  runs  23  t  yards  and  the  hare  runs  21  t  yards.  Since  the 
hound  must  run  79  yards  farther  than  the  hare,  Ave  have  '2Zt  =  21  t  -{•  79. 

The  data  involved  in  this  problem  may  be  represented  con- 
veniently in  the  following  form  : 


Eate 

Time 

DiSTANCB 

Hound  

Hare 

23 
21 

t 
t 

23  f 

2U  +  79 

5.  An  ocean  liner  making  21  knots  an  hour  leaves  port  when 
a  freight  boat  making  8  knots  an  hour  is  already  1210  knots 
out.     In  how  long  a  time  will  the  liner  overtake  the  freight? 


Rate 

Time 

Distance 

Liner 

Freight  boat 

21 

8 

t 

t 

2\t 
8  t  +  1240 

Make  a  similar  diagram  for  each  of  the  following  problems  • 

6.  A  motor  boat  starts  7  miles  behind  a  sailboat  and  runs 
13  miles  per  hour  while  the  sailboat  makes  6  miles  per  hour. 
How  long  will  it  require  the  motor  boat  to  overtake  the  sail- 
boat ? 

7.  A  freight  train  running  25  miles  an  liour  is  200  miles 
ahead  of  an  express  train  running  45  miles  an  hour.  How  long 
before  the  express  will  overtake  the  freight  ? 

8.  A  bicyclist  averaging  12  miles  an  hour  is  52  miles  ahead 
of  an  automobile  running  20  miles  an  hour.  How  soon  will  the 
automobile  overtake  him  ? 


PROBLEMS   INVOLVING   MOTION  201 

9.  A  and  B  run  a  mile  race,  A  runs  18  feet  per  second 
and  B  17. V  feet  per  second.  B  has  a  start  of  30  yards.  In 
how  many  seconds  will  A  overtake  B  ?  Which  will  win  the 
race? 

If  in  each  of  the  examples  4  to  9  we  call  the  rate  of  the 
faster  moving  object  r^  (read  r  one)  and  that  of  the  slower  rg 
(read  ?'  two)f  then  the  distance  traveled  by  the  first  in  the  re- 
quired time  t  is  7\t,  and  that  traveled  by  the  second  is  rjt 

Then  if  n  is  the  distance  which  the  first  must  go  in  order 
to  overtake  the  second,  we  have  the  second  formula  for  motion  : 

r^t=r4-\-n.  (2) 

The  solution  of  (2)  for  t  gives  the  time  required  in  each  problem  for 

the  tirst  to  overtake  the  second- 
Equation  (2)  summarizes  the  solution  of  all  problems  like  those  from 

4  to  9. 

It  is  important  that  the  formulas  for  motion  problems,  (1^ 
on  page  199,  and    (2)  just  derived,  should  be  clearly  under 
stood,  since  they  are  constantly  used  in  solving  problems  of 
this  kind. 

10.  A  fleet,  making  11  knots  per  hour,  is  1210  knots  from 
port  when  a  cruiser,  making  19  knots  per  houi",  starts  out  to 
overtake  it.     How  long  will  it  require  ? 

Use  formula  (2). 

11.  In  how  many  minutes  does  the  minute  hand  of  a  clock 
gain  15  minute  spaces  on  the  hour  hand  ? 

Using  one  minute  space  for  the  unit  of  distance  and  1  minute  as 
the  unit  of  time,  the  rates  are  1  and  yV  respectively,  since  the  hour  hand 
goes  3*^  of  a  minute  space  in  1  minute.  Letting  t  be  the  number  of 
minutes  reqiured,  we  have,  using  formula  (2),  1  •  t  =  ^  t  +  15. 

12.  In  how  many  minutes  after  4  o'clock  will  the  hour  and 
minute  hands  be  together  ?  (Here  the  minute  hand  must  gain 
20  minute  spaces.)     Ans.  21^  min. 


202  LITERAL   EQUATIONS   AND   THEIR   USES 

13.  At  what  time  between  5  and  6  o'clock  is  the  minute 
hand  15  minute  spaces  behind  the  hour  hand  ?  At  what  time 
is  it  15  minute  spaces  ahead  ? 

Since,  at  5  o'clock,  it  is  25  minute  spaces  behind  the  hour  hand,  in 
the  first  case  it  must  gain  25  —  15  =  10  minute  spaces,  and  in  the 
second  case  it  must  gain  25  +  15  =  40  minute  spaces.  Make  a  dia- 
gram as  in  the  preceding  problem  to  show  both  cases. 

14.  At  what  time  between  9  and  10  o'clock  is  the  minute 
hand  of  a  clock  30  minute  spaces  behind  the  hour  hand  ?  At 
what  time  are  they  together? 

15.  A  fast  freight  leaves  Chicago  for  New  York  at  8  :  30  a,m., 

averaging  32  miles  per  hour.     At  2 :  30  p.m.  a  limited  express 

leaves  Chicago  over  the  same  road,  averaging  5d  miles  per  hour. 

In  how  many  hours  will  the  express  overtake  the  freight  ? 

If  the  express  requires  t  hours  to  overtake  the  freight,  the  latter 
had  been  on  the  way  t  +  6  hours.  Then  the  distance  covered  by  the 
express  is  55  t,  and  the  distance  covered  by  the  freight  is  32  (?  +  6). 
As  these  must  be  equal,  we  have  56  t  =  S2  {t  +  6). 

16.  In  a  bicycle  road  race  one  rider  averages  191  miles  per 
hour,  while  another,  starting  40  minutes  later,  averages  22J  miles 
per  hour.    In  how  long  a  time  will  the  latter  overtake  the  former  ? 

III.  PROBLEMS  INVOLVING  THE  LEVER 

Two  boys,  A  and  B,  play  at  seesaw.  They  find  that  the 
teeter  board  will  balance  when  equal  products  are  obtained  by 
multiplying  the  weight  of  each  by  his  distance  from  the  point 
of  support. 

^(100  lbs.)  B(so  lbs.) 

] ^ J 

4  feet  5  feet. 

Thus,  if  B  weighs  80  pounds  and  is  5  feet  from  the  point  of  support, 
then  A,  who  weighs  100  pounds,  must  be  4  feet  from  this  point,  since 
80  X  5  =  100  X  4. 

The  teeter  board  is  a  certain  kind  of  lever  ;  the  point  of 
support  is  called  the  fulcrum. 


PROBLEMS  INVOLVING  THE  LEVER        203 

In  each  of  the  following  problems  make  a  diagram  similar 
to  the  figure  on  the  opposite  page : 

lo  A  and  B  weigh  90  and  105  pounds  respectively.  If  A  is 
seated  7  feet  from  the  fulcrum,  how  far  is  B  from  that  point? 

2.  Using  the  same  weights  as  in  the  preceding  problem,  if 
B  is  6i-  feet  from  the  fulcrum,  how  far  is  A  from  that  point  ? 

3.  A  and  B  are  5  and  7  feet  respectively  from  the  fulcrum 
If  B  weighs  75  pounds,  how  much  does  A  weigh  ? 

4.  A  and  B  weigh  J  00  and  110  pounds  respectively.  A 
places  a  stone  on  the  board  with  him  so  that  they  balance 
when  B  is  6  feet  from  the  fulcrum  and  A  SJ-  feet  from  this 
point.     How  heavy  is  the  stone? 

Formula  for  Lever  Problems.  If  the  distances  from  the  boys 
to  the  fulcrum  are  respectively  di  and  dg,  and  their  weights  w^ 
and  W2,  then 

This  is  our  formula  for  solving  lever  problems. 

This  equation  is  a  statement  in  the  language  of  algebra  of  a  very 
important  law  of  nature.  The  law  is  the  result  of  a  very  large  num- 
ber of  careful  experiments.  It  is  a  universal  custom  among  scientific 
men,  to  express  laws  of  nature,  so  far  as  possible,  by  means  of  literal 
equations  of  this  sort. 

If  any  three  of  the  four  numbers  di,  w\^  d_>,  w^,  are  given,  the  fourth 
may  be  found  by  means  of  the  equation  d^wi  =  d-2,W2- 

5.  Solve  d^W]^  =  d^io.y  for  d^  in  terms  of  the  other  three  letters. 
Then  solve  for  iD]^. 

6.  Solve  dxW^  =  d2^02  for  d^  and  also  for  w^. 

7.  A  and  B  are  seated  at  the  opposite  ends  of  a  13-foot 
teeter  board.  Using  the  weights  of  problem  1,  where  must  the 
fulcrum  be  located  so  that  they  shall  balance  ? 

If  the  fulcrum  is  the  distance  d  from  A,  then  it  is  (13  —  d)  from  B 
Hence,  90  d  =  105(13- d). 


204  LITERAL   EQUATIONS  AND   THEIR   USES 

8.  A,  who  weighs  75  pounds,  sits  7  feet  from  the  fulcrum. 
If  B  weighs  105  pounds,  at  what  distance  from  the  fulcrum 
should  he  sit  in  order  to  make  a  balance  ? 

9.  A  and  B  together  weigh  212^  pounds.  They  balance 
when  A  is  6  feet,  and  B  6J  feet,  from  the  fulcrum.  Find  the 
weight  of  each. 

10.  A  lever  9  feet  long  carries  weights  of  17  and  32  pounds 
at  its  ends.  Where  should  the  fulcrum  be  placed  so  as  to  make 
the  lever  balance  ? 

11.  A  lever  of  unknown  length  is  balanced  when  weights  of 

30  and  45  pounds  are  placed  on  it  at  opposite  ends.     Find  the 

length  of  the  lever,  if  the  smaller  weight  is  two  feet  farther 

from  the  fulcrum  than  the  greater. 

Suggestion.  Let  x  be  the  distance  from  the  greater  weight  to  the 
fulcrum. 

IV„  MISCELLANEOUS  LITERAL  EQUATIONS 

Solve  the  followinsr  for  each  letter  in  terms  of  the  others : 


'& 


1.   i^=32H-^  a 


2.    Z  =  a  +  (?i  —  1)  d. 


Solve  each  of  the  following  for  x : 

5.    ax -\- 3  b  =  ex -}- d.  _ .     x  ,  x      x      ^ 

11.    — r  T    I —  —  -•- 


a      b      0 

a  -\-  bx  _  c  -\-  dx 
a  -\-b        c  -\-  d 
3^    ax-\-b_bx-\-c^^^  ^^^^      ^_^ 


6.  (a  —  x)  (b  +  x)  =  x{b — x). 

7.  {x-^a){x-\-b)  =  {x—cy'  12. 


c  d 


13. 


1  -\-x     1—x 


-     X  ,      X  a  ,  .      3  a.T       o 

9.    -H ■  = 14. —2a  =  i^x. 

b      a  —  b      a  -\-  b  a  —  b 

-^aj  +  l      a-\-  b  -_aic— 1      1  +  bx  .  ^ 

lU.    —  =  •  15.     — '■ = r  '^• 

x  —  1      a—  b  bx  ax 


MISCELLANEOUS   LITERAL   EQUATIONS  205 

X  —  a     X  -\-  a _    2x  a-\-x     b+x     c-^x_^ 

a— ba-\-ba-^b  a  b  c 

18     a;  +  l      x-1^     2x(x-h2) 
x-1      a;  +  l      (x  +  l){x-l) 

x-j-m      x  —  m  _      2  x(x  +  1) 
x  —  m     x  +  m      (x -\- m)(x  —  m) 

20    ^'^^  4-  ^~  ^  =      ^ ^(^  +  a) 
x  —  a      x-{-a      (x-\-a){x  —  a) 

21.  What  number  must  be  added  to  each  term  of  the  frac- 
tion 4  to  obtain  a  fraction  equal  to  i  ? 

22.  What  number  must  be  added  to  each  term  of  the  fraction 

(X  .  .  c 

-  to  obtain  a  fraction  equal  to  -? 
b  d 

Solve  Example  21  by  substituting  in  the  formula  obtained  under  Ex- 
ample 22. 

23.  What  number  must  be  subtracted  from  each  term  of  the 
fraction  -^^  to  obtain  a  fraction  equal  to  f^? 

24.  What  number  must  be  subtracted  from  each  term  of  the 

'  CI  •  •  c 

fraction  -  to  obtain  a  fraction  equal  to  -? 
b  ^  d 

Solve  Example  23  by  substituting  in  the  formula  obtained  under  Ex- 
ample 24. 

25.  What  number  must  be  added  to  each  of  the  numbers  2, 
5,  4,  12  so  that  the  sums  shall  be  in  proportion  when  taken  in 
the  given  order  ? 

26.  AVhat  number  must  be  added  to  each  of  the  numbers  a, 
b,  c,  d  so  that  the  resulting  sums  shall  be  in  proportion  when 
taken  in  the  given  order  ? 

27.  What  number  must  be  subtracted  from  each  of  the 
numbers  a,  b,  c,  d  so  that  the  resulting  remainders  shall  be  in 
proportion  when  taken  in  the  given  order  ? 

Compare  the  results  in  Examples  26  and  27  and  explain  the  relation 
between  them. 


206  LITERAL   EQUATIONS   AND   THEIR   USES 

HISTORICAL   NOTE 

Literal  Equations  whose  solutions  lead  to  formulas  belong  to  a  very 
late  stage  in  the  development  of  algebra.  In  the  earliest  attempts 
to  solve  equations,  the  unknown  was  represented  by  some  word  such  as 
res,  the  thing  (see  page  41).  Later  a  single  letter  was  used  for  the  un- 
known, but  known  quantities  were  still  represented  by  numerals,  and  the 
equations  were  so-called  numerical  equations.  Finally,  both  known  and 
unknown  quantities  were  represented  by  letters,  and  such  equations  were 
called  literal  equations.  Numerical  equations  lead  only  to  special  solu- 
tions, while  literal  equations  lead  to  general  solutions  or  formulas. 

The  Greeks  solved  special  cases  of  general  problems  (see  page  229) ,  but 
never  obtained  general  formulas.  Vieta  (see  page  101)  used  letters,  as 
we  do,  to  represent  both  known  and  unknown  quantities,  but  he  always 
used  capital  letters.  Harriot  (see  page  101),  following  Vieta,  likewise 
used  letters,  but  usually  small  letters  instead  of  capitals.  Newton  (see 
page  62)  was  the  first  to  let  a  letter  stand  for  negative  as  well  as  posi- 
tive numbers.     For  further  uses  of  literal  equations  see  page  217. 

REVIEW   QUESTIONS 

1.  State  the  rules  which  may  be  derived  from  the  formula 

V  =  Iwh, 
where  these  letters  represent  the  volume,  length,  width,  and 
height,  respectively,  of  a  rectangular  solid. 

2.  State  all  the  rules  of  interest  which  may  be  derived  from 
the  formula  /  =  prt. 

3.  State  all  the  rules  of  interest  which  may  be  derived  from 
the  formula  a  =  p-\- prt. 

4.  Which  problems  on  motion  in  this  chapter  can  be  solved 

by  use  of  the  formula 

d=ri? 

5.  Which  problems  on  motion  in  this  chapter  belong  to  the 
class  whose  solutions  are  summarized  by  the  solutions  of  the 
equation  r^i  =  r.>i  -\-  n  ? 

6.  State  fully  the  meaning  of  the  equation  Wid^  =  w^d^  iu 
connection  with  the  lever. 


CHAPTER   XIV 

SIMULTANEOUS  EQUATIONS   OF  THE   FIRST  DEGREE 

147.  An  Equation  of  the  First  Degree.  An  equation  of  the 
first  degree  in  x  and  y  is  one  in  which  neither  x  nor  y  is  multi- 
plied by  itself  or  by  the  other.  For  instance,  such  an  equation 
must  not  contain  x"^,  y^,  or  xy. 

E.g.  13  aj  —  5  2/  =  14  is  of  the  first  degree  in  x  and  y,  while  2xy  —  x  =  b 
and  3  a;  —  5  2/2  =z  13  are  not  of  the  first  degree  in  x  and  y. 

Indeterminate  Equations.  A  single  equation  in  two  unknowns 
is  satisfied  by  indefinitely  many  pairs  of  values  of  the  un- 
knowns.    Such  an  equation  is  called  an  indeterminate  equation. 

E.g.     x  +  2/  =  5  is  satisfied  by 

x  =  ll      x  =  '2,\     x  =  01      x——\ 


y  =  ^]'   y  =  Z\'   y  =  5j'   y  =Q      ''   "'"• 

Simultaneous  Equations.    Two   equations  in  two  unknowns 

which  are  satisfied  by  the  same  pair  of  values  of  the  unknowns 

are  called  simultaneous  equations. 

E.g.  X  -\-y  =  b  and  x  ~  y  =  S  are  simultaneous,  because  both  are 
satisfied  hj  x  =  4,  y  =  1. 

Contradictory  Equations.  Two  equations  in  two  unknowns, 
which  cannot  be  satisfied  by  the  same  pair  of  values  of  the  un- 
knowns, are  called  contradictory  equations. 

E.g.  X  +  y  =  b  and  x  -\-  y  =  2  are  contradictory,  since  no  pair  of  values 
of  X  and  y  can  make  their  sum  both  5  and  2  at  the  same  time. 

Dependent  Equations.  Two  equations  in  two  unknowns  are 
said  to  be  dependent  if  one  can  be  derived  from  the  other.  In 
this  case  every  pair  of  values  which  satisfies  one  will  also  satisfy 
the  other. 

E.g.  X  -\-  y  =  b  and  2  a--  +  2  ?/  =  10  are  dependent,  since  the  second 
may  be  derived  from  the  first  by  multiplying  both  members  by  2. 

207 


208       SIMULTANEOUS  EQUATIONS   OF  THE  FIRST   DEGREE 

Independent  Equations.     Two  equations  in  two  unknowns  are 
independent   if  neither  can  be  derived  from  the  other. 
E.g.     X  -{-  y  =  ^  and  x—  y  =  3  are  independent  equations. 
Summary. 

The  equations   '  o  r  4-  2  w  —  lO     ^^^  simultaneous  but  not  inde- 
pendent. 

IX  4-  y  :=  ^  ] 
I      —of  ^^®    independent    but   not   simul- 
taneous. 

Z  Q  r  ^^'^  both  simultaneous  and  iwde- 
pendent. 

If  two  equations  of  the  Hrst  degree  in  x  and  y  are  both  simul- 
taneous and  independent,  they  have  one  and  only  one  pair  of 
values  of  x  and  y  which  satisfies  both. 

E.g.  x  +  y  =  5  and  x  —  y  =S  are  simultaneous  and  independent,  and 
they  are  satisfied  hy  x  =  i,  y  =  1  and  by  no  other  pair  of  values  of  x  and  y. 

148.  Elimination.  To  solve  two  simultaneous  independent 
equations  of  the  iirst  degree  in  x  and  y  is  to  find  the  one  pair 
of  values  of  x  and  y  which  satisfies  both.  In  this  chapter  such 
equations  will  be  solved  by  the  algebraic  method  called  elimi- 
nation. The  essential  step  in  elimination  consists  in  combining 
the  equations  in  such  a  way  as  to  get  rid  of  one  of  the  unknown 
numbers. 


(1)  Solve 


ELIMINATION  BY  ADDITION   OR  SUBTRACTION 
149.    Illustrative  Examples. 

x-\-2y  =  7,  (1) 

Sx-4y  =  l.  (2) 

Multiplying  both  members  of  equation  (1)  by  2,  we  have, 

2x-\-4y  =  U.  (3) 

Adding  the  members  of  equations,  (2)  and  (3),  -1-  4y  and  —iy  cancel. 

Hence,  6x  =  15,  and  x  =  8. 

Substituting  in  (1),         3  -\-  2y  z=  7,  and  y  =  2. 

Verify  this  by  substituting  x  =  3,  ?/  =  2,  in  (1)  and  (2). 


ELIMINATION   BY   ADDITION   OR   SUBTRACTION       209 

(2)Solve           rTx  +  3,  =  4.-,  +  10,  (1) 

|3aj-  y=4:y-3x-\-7.  (2) 
Collecting  the  unknowns  in  each  equation,  we  have, 

From  (1),                                3x  +  4t/  =  10,  (3) 

From  (2),                                6  x  -  5  y  =  7.  (4) 

Multiplying  both  members  of  equation  (3)  by  2,  we  have, 

6x  +  Sy  =  20.  (5) 

Subtracting  equation  (4)  from  equation  (5),  we  have, 

13  2/ =  13.  (6) 

By  D  I  13,  y  =  l. 

Substituting  in  (3),  x  =  2. 

Hence,  the  solution  is  x  =  2,  y  =  1. 

The  process  used  in  the  solution  of  example  (1)  is  called 
elimination  by  addition ;  that  used  in  example  (2)  is  called 
elimination  by  subtraction. 

Rule  for  Elimination  by  Addition  or  Subtraction. 

(A)  Whe^^  the  coefficients  of  one  of  the  unhnowns  are 
numerically  equal  in  the  two  equations: 

(1)  If  these  coefficients  have  opposite  signs,  add  the 
equations;  if  they  have  lihe  signs,  suhti^act  them. 

(2)  Solve  the  resulting  equation,  thus  finding  the  valus 
of  one  of  the  unhnowns.  Substitute  this  value  in  one 
of  the  given  equations,  thus  finding  the  other  unknown. 

(3)  Clzech  the  results  hy  substituting  in  both  of  tlie 
given  equations. 

(B)  When  the  coefficients  of  neither  one  of  the  unkivowns 
are  numerically  equal,  make  one  pair  so  by  multiplying 
the  equations  by  suitable  numbers,  and  then  proceed  as 
in  {A). 

Note.  —  We  commonly  say  ''add  two  equations  "  instead  of  using  the 
longer  expression  "add  the  corresponding  members  of  the  equations." 
Similarly  for  subtraction.  Likewise,  we  say  "multiply  the  equation" 
and  "divide  the  equation"  instead  of  "multiply  both  members  of  the 
equation"  and  "  divide  both  members  of  the  equation." 


210       SIMULTANEOUS    EQUATIONS   OF   THE   FIRST   DEGREE 


ORAL  EXERCISES 

Solve  the  following  pairs  of  equations  by  addition  or  sub- 
traction.    Check  each  solution. 


1. 


3. 


x-{-y  =  2, 
X  —  y  =  0. 

x  +  y  =  S, 
[X-y  =  l. 

x-\-y  =  4:, 

[X-y  =  2. 


x-hy  =  5, 
X  —  y  =  1. 

x-^y  =  6, 

yx-y=2. 

2  cc  +  2/  =  3, 
x  —  y  =  0. 


8. 


a^  +  22/  =  5, 
X  +  ?/  =  4. 

2x  +  y  =  10, 
a; +  2/ =  6. 

3a;  +  22/  =  7, 
3  a;  +  2/  =  5. 


2. 


3. 


WRITTEN 

Solve  the  following  pairs  of 

2  a;  +  3  2/  =  22, 
x-y  =  l. 

5x-2y  =  21, 
X  —y  =  6. 

6  a;  +  30  =  8  2/, 
[32/ +  17  =  2 -3a;. 

S  X  —  4,y  =  12  X, 

4x  +  22/  =  3-f-42/. 

x-^6y  z=2  X—  16, 
3x-2y=24:. 

5a;  +  10?y  =  -7, 
2x-\-5y=  —2. 

5x+3y  =  -2, 
'Sx  +  2y  =  -l. 

3  a  +  7  6  =  7, 
1  5  a  +  3  6  =  29. 

r  =  3  5  -  19, 
.9  =  3  r  -  23. 

2p  =  5q-16, 

(7q=-3p-\-5. 


EXERCISES 

equations.     Check  the  first  six. 
^^     f7w  =  27i-3, 


12. 


6. 


8. 


9. 


10. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


19  n  =  6  m  +  89. 

15  /c  =  10  -  20  I, 
25  k -301  =  80. 

6  c  + 15  d  =  —  6, 
21  d  -  8  c  =  -  74. 

I  2  a;  -  3  2/  =  4, 
\2y-3x  =  -21. 

u  -\-v  =  27, 
^v  =  ld-^u. 

7  a  =  1  -\- 10  y, 

16  y  =  10  a-  1. 

28  X  -h  14  2/  =  23, 
14  a; -14  2/ =  1. 

5  a;  +  2  2/  =  a;  -f  18, 
2x-{-3y=3x  +  27. 

7  y  —  x  =  x  —  17, 
2y-h3x=3S. 

i)x  +  2y  =  -2, 
X  —  -ly  =  —  35. 


ELIMINATION   BY   SUBSTITUTION  211 

ELIMINATION   BY   SUBSTITUTION 

150.    Illustrative  Problem.     Solve  the  equations : 

2a: +  3?/ =13.  (1) 

5x-6y  =  -S.  (2) 

From  (1)  3 2/  =  13  -  2  ic,  or  y  =  l^-H-^.  (3) 

o 

2 
Substituting  in  (2) ,      5  x  -  ^(13- 2  a:)  ^  _  g.  (4) 

By  i^,  5  a:  -  26  +  4  a:  =  -  8.  '                                 (5) 

Byl,  ^,                                                   9x=18.  (6) 

ByD,                                                        x  =  2.  (7) 

From  (3),  ^^13-2. 2^3  ^g^ 

Verify  this  by  substituting  these  values  of  x  and  y  in  (1)  and  (2). 

The  process  here  used  is  called  elimination  by  substitution. 

This  process  is  convenient  when  no  fraction  remains  after 
the  substitution.  This  was  the  case  in  the  above  solution  be- 
cause the  coefficient  of  ?/  in  (2)  is  a  multiple  of  that  in  (1). 

Rule  for  Elimination  by  Substitution. 

(1)  Solve  one  equation  for  one  itnhnown  in  terms  of 
the  other  unhnoivn. 

(2)  Substitute  the  expression  for  this  unhnoivn  in  the 
second  equation. 

(3)  Solve  the  resulting  equation,  thus  finding  the  value 
of  one  unhnown. 

(4)  Substitute  this  value  in  one  of  the  given  equations, 
thus  finding  the  second  unhnown. 

(5)  Chech  by  substituting  the  results  in  both  of  the 
given  equations. 

ORAL  EXERCISES 

1.  If  X  =  6,  what  is  the  value  of  y  in  2x-\-  y  =  16? 

2.  If  ic  =  4,  what  is  the  value  of  y  in  x  -\-3y  =  16? 

3.  If  a;  =  y,  what  is  the  value  of  2/  in  x  -i-2y  =  6? 

4.  If  .1*  =  y,  what  is  the  value  of  .x*  in  3  a;  +  2  ?/  =  15  ? 
6.  If  a;  =  2 y,  what  is  the  value  of^  in  x-^2y  =  S? 


212      SIMULTANEOUS  EQUATIONS   OF  THE   FIRST  DEGREE 

WRITTEN  EXERCISES 

Solve  the  following  by  substitution : 

J  (x-^2y  =  4,  g  (x-y  =  S7, 

[2x-\-y  =  5.  '    \2x-\-3y=:31x-^13y. 

2  (3x-y  =  5,  ^      { 2x  —  y  =  y-\-6, 
\DX  +  2y  =  23.  '     ^  +  2^  =  42/4-3. 

3  (2x  +  y  =  3,  (5x-^3y  =  0, 
[3x-7y  =  30.                       '    {2x-\-y  =  l. 

5y  +  x  =  7,  g  {3x-2y  =  3, 


5x-3y  =  4:-2x~\-7.  (2x -}-3y  =  6x-l. 

ox-\-Sy=—l,  ^^      ( 5x  —  3y  =  0, 

Qy  —  x  =  4:y  —  7.  [2x  —  6y  =  —  x. 


WRITTEN  EXERCISES 

Solve  by  either  method  of  elimination : 

^      lx  +  y  =  ^,  ^  i2x-{-3y  =  5, 

\x-y  =  10.  '  \6x  +  14:y  =  0. 

2      lx-y=^-3,  ^^  Ux-\-3y  =  5, 

(x  +  4?/=12.  *  \7x-2y  =  74:. 

^      f2x-\-3y  =  5,  ^^  l6y-h2x  =  U, 

\7x-5y  =  33.  '  \3y -\-12x=lS. 

[4a;  +  32/  =  — 6.  '  \x-{-y  =  —  o. 

[2a;-4?/  =  8,  i  3  f/ -{- 5x  =  12 -^2x, 

\3x-^2y  =  4.  [17x-y  =  4:y-20. 

f  3  a;  _  4  ^y  =  8,  f  6  +  rr  +  ?/  =  2  .r  -  1 , 

6.  14. 

I  2a; +  3  2/ =  11.  \3y  +  x=:6  y -\-9. 

^      Uy-2x  =  2,  ^^  \y-]-r>x=2x-i-5, 


2y-^5x  =  7.  '     \2y-3x^l9. 

3x-y  =  2x-l,  i6x  +  2y  =  22, 

12x +  y  =  U.  ^^'     [10x-5y=20. 


FRACTIONAL  EQUATIONS 


213 


17. 


18. 


19. 


20. 


21. 


5  a;  +  4  2/ =  11. 

7x-4.y  =  3, 
5x  -\-  Sy  =  5. 

12?/- 10a;  =  -6, 

7y-\-x  =  99. 

7x  —  3y  =  —  7, 
5y-9x=l. 

7x+4:y  =  S, 
2x-\-3y  =  25. 


22. 


23. 


24. 


25. 


26. 


[  34  a;  +  TO  ?/  =  4, 
[5x-Sy  =  -36. 

I  7  X  4-  9  ?/  =  8, 
[2x-Sy  =  19. 

[  8  a;  +  4  ?/  =  49, 
[5a;-8^=28. 

I  4  a;  +  7  2/  =  7, 
(  5  a;  —  2  y  =  41. 

r  8  x  -h  4  7/  =  -  28. 
3  a; +  9?/  =  12. 


SIMULTANEOUS   FRACTIONAL    EQUATIONS 

151.  Reduction  to  Standard  Form.  The  equations  thus  far 
given  have  for  the  most  part  been  written  in  a  standard  form, 
ax  -{-  by  =  c,  in  which  all  the  terms  containing  x  are  collected, 
likewise  those  containing  y,  and  those  which  contain  neither 
X  nor  y.  When  the  equations  are  not  given  in  this  form, 
they  should  be  reduced  to  this  form  at  the  outset,  as  in  the 
following  solution : 


Example.     Solve 

\7y-4.      2x-3_, 
5        '        2      -'' 
5x-2     2y  +  l      o 
3                5 

(1) 

(2) 

Solution.     (1)  X  10, 

14y  _8  +  lOo:-  15  =  15. 

(3) 

Transposing  in  (3), 

Uy  -\-  10x  =  38. 

(4) 

(4)  ^  2, 

7y+5x  =  19. 

(5) 

(2)  X  15, 

25a; -10 +  6?/ +  3:=  30. 

(6) 

Transposing  in  (6) , 

25x  +  6y  =37. 

(7) 

(5)  X  5, 

25  a;  +  35?/  =  95. 

(8; 

(8)-(7), 

29?/ =  58. 

(9) 

(9)  -^  29, 

y  =  2. 

(10) 

Substituting  in  (7), 

a:=  1. 

Check.     Substitute  x  = 

:  1,  ?/  =  2  in  (1)  and  (2)  and  see 

that  each  is 

satisfied. 

214      SIMULTANEOUS   EQUATIONS   OF  THE   FIRST   DEGREE 


EXERCISES 

After  reducing  each  of  the  following  pairs  of  equations  to 
the  standard  form,  solve  them  by  means  of  either  process  of 
elimination : 


7x-15 

2  X  —  y  =  S. 

5y 

x-\-7y  =  6. 

\x-hy  ,x-y_.f. 
3.     '      2     "^     2      ~      ' 
2x-y  =  16. 


2x-y     3y-x_^^^ 


10. 


7m-f8      ln  —  1  _  _  9     11. 

5       ~      4 
2m  —  4      ?i  —  1  _      j^ 


~4~  +  ~^-^^  . 
x+1      2y-4^^ 

2  7 

[8  a -3,  55-2     .„ 
9  3 

2a  +  7      35  +  10 
10 


[7y-4      2a;-3_ 
5      "^       2      ~    ' 
6a;-3      2?/4-l^j 

5  5 

^ns.  X  =  5y3-,  2/  =  2if . 


3.7  +  7      5a;-7^ 

2  3 

2ic-4      2?/-l_ 


10, 


01 


5+3p     5g-2^  _  9 
74"' 
6p  +  8g  =  108. 

{Sx-2y  =  4:, 
2a;-l      7y-4^-^g 


—  _  ^4 


15. 


2  a;  —  ?/      3_a;  —  y  _o 
~^~'^       3    ■  ~   ' 
5x  -{-y     9x  —  2y  _  ^ 
10     ~        5        ~' 


12. 


13. 


14. 


^MS. 


f  5.x'  +  7?/  =  89i 
'^  a;  —  4  ,  6  ?/ —  1  _  ^ ^i 
5 


+  "-^h-^  =  13^. 


[32  a; -9?/ =  299, 
2^j--5 _ 3j_— J- _ _ -j^g 

7  2 

5  a;  -  12  ?/  =  4, 

2a;-7      3?/ -4^      ^ 
34"" 


X  =  1000, 
2/  =  2610. 


FRACTIONAL   EQUATIONS 


215 


152.    Equations  with  Literal  Denominators. 
So]  ve  the  equations  : 

4,6  36 


x-y 
3 


+ 


x-\-y 

2 


,2   _    yl 


-  18 


2x-y      x-3y      {2x  —  y){x  —  3y) 

Multiplying  (1)  by  the  L.  C.  D.,  x^  _  ^2^ 

4(x  +  y)+6(x-y)=S6. 
By  F,  D,  ox-y  =  18. 

Multiplying  (2)  by  the  L.  CD.,  (2x  -y){x-3  y), 

3(x-3y)-2{2x-y)  =  -  18. 
By  F,  A  x+ly  =  18. 

Multiplying  (4)  by  7,  Zbx-ly  =  126. 

Adding  (6)  and  (7),  36  a:  =  144. 

By  i>,  x  =  4. 

Substituting,  x  =  4  in  (0),  y  =  2. 

Check  by  substituting  x  =  4,  ?/  =  2  in  (1)  and  (2). 


(1) 
(2) 


(3) 
(4) 

(5) 
(6) 
C7) 
(8) 
(9) 
(10) 


3. 


8a^  +  24y^3 
y-2x 

""-^y  =17. 

x-{-2y  +  2 

2>x-\-2  ^x  +  l 
Sy  —  o     y  —  l' 
3x-2     3a.'-l 


X 


_    2y 


x-^1      2y-3' 
I  a;  —  1      y-\-2 


y-hl        y-1       (^-1)(^+1) 
2g;  —  1  _  3  ?/  —  1  _  —  xy 


x  +  1         y-\.l       (a;  +  l)(2/  +  iy 


x-\-2    ,  2a;-l 


5  .TV 


2y-l       2/  +  1 

a;-3      a;  +  2_ 

y-2     2/4-1"  (2/  +  1)(2/  -  2)  ' 
2  a;  +  1      .T  +  o  7 


(22/-l)(2/  +  l) 
2 


2  2/ +  5      .7/4-4      (2  2/4-5X2/4-4)* 


216      SIMULTANEOUS   EQUATIONS   OF   THE   FIRST   DEGREE 

153.    Special  Case.     In  examples  like  the  following  it  is  best 
not  to  reduce  the  equations  to  the  integral  form. 

Example.     Solve  the  equations  : 


'^  +  2  =  2, 
X     y 

20-21  =  3. 

X 


y 


Solve  the  equations  for    -  and  -  instead  of  for  x  and  y. 

y 


Multiplying  (1)  by  7, 

Adding  (2)  and  (3), 

Hence  by  Z>, 

Substituting  -  =  -  in  (1), 
X      2 

From  (5)  and  (6)  by  M, 


X 


X       y 


X 
1 
X 
1 

y 


x  =  2,  y  =  S. 


(1) 
(2) 

(3) 
(4) 
(5) 

(6) 
(7) 


Tr}^  to  solve  these  equations  by  first  clearing  of  fractions. 


WRITTEN  EXERCISES 

Solve  the  following  equations  : 


2. 


1  +  1=4, 

?  +  l=21. 

[12      10      . 

X    y               ^ 

X     y 

7. 

X        y 

1-1  =  2. 

i-2  =  -19. 

-9  +  2  =  15. 

X     y 

X     y 

X     y 

1-I  =  -13, 

I_^_191 

|'^+?=i, 

^    y 

a      0 

X     y 

5. 

8. 

-  +  ?  =  12. 

2  +  12  =  24. 

1+5=1. 

[x      y 

a       h 

X      y 

(5-5  =  2, 

^--  =  -4, 

-  +  -  =  20, 

t        V 

X'         ?/ 

X     y 

6. 

9. 

11  +  1  =  67. 

«  +  H  =  52. 

i-l  =  30 

.    '"            t 

X       y 

X     y 

LITERAL    EQUATIONS 6  217 

SIMULTANEOUS  LITERAL  EQUATIONS 

154.    Examples.     1.    The  sum  of  two  numbers  is  35  and  their 

difference  is  5.     What  are  the  numbers  ? 

Let  X  represent  one  number  and  y  the  other.  Form  two  equations  and 
solve  them. 

2.  The  sum  of  two  numbers  is  48  and  their  difference  is  24. 
What  are  the  numbers  ? 

3.  The  sum  of  two  numbers  is  41^  and  their  difference  is 
23^.     What  are  the  numbers  ? 

4.  The  sum  of  two  numbers  is  8590  and  their  difference  is 
3480.     What  are  the  numbers  ? 

5.  If  the  sum  of  two  numbers  is  s  and  their  difference  is  rZ, 

find  the  numbers. 

Solution.     Let  x  represent  one  of  the  numbers  and  y  the  other. 
Then,  x  +  y  =  s,  (1) 

x-y  =  d.  (2) 

Solving,  we  get  x  =  '-±^  =  |  +  f ' 

and  y=izii?^i_^. 

^  2         2      2 

Translated  into  words  these  results  are : 

One  of  two  numbers  is  equal  to  half  their  sum  plus  half  their 
difference,  and  the  other  is  equal  to  half  their  sum  minus  half 
their  difference. 

6.  Test  this  general  solution  by  applying  it  to  the  following : 

s  =  48.  '  (s  =  8590.  fs  =  40.  fs  =  38, 

cZ=24'  U=348  '  lcZ  =  52'  UZ=50. 

7.  Apply  the  general  solution  to  the  following : 
=  3  fs  =  3^.  fs  =  3i  fs  =  5|, 


f.s  =  f.  {s  =  3i,  1^  =  31.. 

U  =  l'  \d=ll'  \d=n' 


This  shows  again  how  the  solution  of  literal  equations  leads 
to  formulas.     See  pages  196  to  206. 


218      SIMULTANEOUS  EQUATIONS   OF  THE   FIRST   DEGREE 


WRITTEN  EXERCISES 

In  the  following  x  and  y  are  the  unknowns.     Solve  for  them 
in  terms  of  the  other  letters. 


2. 


ax  +  by  = 

=  1, 

\     ex  — 

-  dy  = 

:1. 

2x- 

Sy  = 

c  —  d, 

3  X- 

2y  = 

c-\-d. 

mx  + 

2  ny  : 

=  lc, 

[3  +  2 

7nx  = 

+  ny. 

'  x  +  y 

=  a, 

X      y 
a      b 

=  1. 

a      b_ 

'^x     y 

=  1, 

b+1 

\x      y 

=  1. 

6. 


7. 


1 

.  1 

+  -  = 

a, 

x 

y 

1 

i_ 

b. 

X 

y~ 

a 

,  a 

+  -  = 

■m, 

X 

y 

b 

_b_ 

n. 

X 

y 

a 

.  b 

+  -  = 

ry 

X 

y 

c 

.  d 

+  -  = 

:S. 

X 

y 

PROBLEMS  LEADING  TO  FRACTIONAL  EQUATIONS 

In  solving  the  following  problems  use  two  equations  if  two 
unknowns  are  involved. 

1.  Find  two  numbers  whose  sum  is  51,  such  that  if  the 
greater  is  divided  by  their  difference,  the  quotient  is  3J. 

2.  Find  two  numbers  whose  sum  is  91,  such  that  if  the 
greater  is  divided  by  their  difference,  the  quotient  is  7. 

3  There  are  two  numbers  whose  sum  is  s,  such  that  if  the 
greater  is  divided  by  their  difference,  the  quotient  is  q.  Find 
an  expression  in  terms  of  s  and  q  representing  each  number. 
Solve  Examples  1  and  2  by  substituting  in  this  formula. 

4.  There  are  two  numbers  whose  difference  is  153.  If  their 
sum  is  divided  by  the  smaller,  the  (quotient  is  equal  to  ^.  Find 
the  numbers. 


PROBLEMS   LEADING  TO   FRACTIONAL  EQUATIONS      219 

5.  There  are  two  numbers  whose  difference  is  d.  If  their 
sum  is  divided  by  the  smaller,  the  quotient  is  q.  Find  the 
numbers.     Solve  Example  4  by  substituting  in  this  formula. 

6.  Find  two  numbers  whose  difference  is  320,  such  that  the 
greater  divided  by  their  sum  is  |. 

7.  Find  two  numbers  whose  difference  is  60,  such  that  the 
greater  divided  by  their  sum  is  J. 

8.  Find  two  numbers  whose  difference  is  \,  such  that  the 
greater  divided  by  their  sum  is  |. 

9.  Find  two  numbers  whose  difference  is  d,  such  that  the 

greater  divided  by  their  sum  is  -  • 

h 

10.  A  number  has  two  digits  whose  sum  is  s.  If  the  number 
is  divided  by  the  difference  between  the  digits,  the  quotient  is 
q.     Find  the  number,  the  tens'  digit  being  the  larger. 

11.  There  is  a  number  composed  of  two  digits  whose  sum  is 
11.  If  the  number  is  divided  by  the  difference  between  the 
digits,  the  quotient  is  16|.  Find  the  number,  the  tens'  digit 
being  the  larger. 

155.  Many  problems  may  be  solved  in  two  ways,  namely,  by 
using  one  unknown,  or  by  using  two  unknowns. 

Example.  Find  two  numbers  whose  sum  is  20,  such  that 
when  one  of  them  is  subtracted  from  twice  the  other,  the  re- 
mainder is  16. 

(a)  Using  one  unknown.  Let  a*  represent  one  number.  Then  20  —  x 
is  the  other  number,  and  the  equation  is  2  a:  —  (20  —  a:)  =  16. 

(&)    Using  two  unknowns.     Let  x  and  y  represent  the  two  numbers. 

Tlien,  I    ^  +  ?/  =  20, 

\2x-y  =  \Q. 

The  translation  of  problems  into  equations  is  usually  easier 
when  more  than  one  unknown  is  permitted.  This  is  due  to 
the  fact  that  in  this  case  each  of  the  given  relations  between 
the  numbers  is  put  down  as  a  separate  equation. 


p  =  2l  +  2w 


220      SIMULTANEOUS   EQUATIONS   OF   THE   FIRST  DEGREE 

PROBLEMS  INVOLVING  TWO  UNKNOWNS 
1.    If  66  ana  I  are   the  width   and  length  of  a  rectangle, 

I — 1      express  its  perimeter  in  terms  of  w  and  I. 

K,    Also  its  area. 

I  2.    If  the  width  of  the  rectangle  in  the 

preceding  is  increased  by  10  and  its  length 
by  20,  express  its  new  perimeter  and  also  its  new  area  in  terms 
of  IV  and  I. 

3.  If  X  and  1/  represent  the  ages  of  a  father  and  son  respec- 
tively, represent  the  sum  of  their  ages  5  years  ago  in  terms  of 
X  and  y. 

4.  If  a  number  consisting  of  two  digits  is  increased  by  15 
by  changing  the  order  of  its  digits,  which  is  greater,  the  digit 
in  tens'  or  in  units'  place  r 

5.  If  a  number  consisting  of  two  digits  is  decreased  by 
changing  the  order  of  its  digits,  which  is  greater,  the  digit  in 
tens'  or  in  units'  place  ? 

6.  A  rectangular  field  is  o2  rods  longer  than  it  is  wide. 
The  length  of  the  fence  around  it  is  308  rods.  , 

Find  the  dimensions  of  the  field. 

7.  Find  two  numbers  such  that  7  times 
the  first  plus  4  times  the  second  equals  37, 
while  3  times  the  first  plus  9  times  the  second  equals  45. 

8.  A  certain  sum  of  money  was  invested  at  5  %  interest 
and  another  sum  at  6  %,  the  two  investments  yielding  $  980 
per  annum.  If  the  first  sum  had  been  invested  at  6%  and 
the  second  at  5  %,  the  annual  income  would  be  $  1000.  Find 
each  sum  invested. 

Suggestion.     Let  x  and  y  represent  the  sums  invested. 


Then 


PROBLEMS   INVOLVING   TWO    UNKNOWNS  221 

9.  The  combined  distance  from  the  sun  to  Jupiter  and 
from  the  sun  to  Saturn  is  1369  million  miles.  Saturn  is  403 
million  miles  farther  from  the  sun  than  Jupiter.  Find  the 
distance  from  the  sun  to  each  planet. 

10.  Find  two  numbers  such  that  7  times  the  first  plus  9 
times  the  second  equals  116,  and  8  times  the  first  minus  4 
times  the  second  equals  4. 

li.  The  sum  of  two  numbers  is  108.  8  times  one  of  the  num- 
bers is  9  greater  than  the  other  number.     Find  the  numbers. 

12.  Two  investments  of  $24,000  and  S  16,000  respectively 
yield  a  combined  income  of  S840.  The  rate  of  interest  on 
the  larger  investment  is  1  %  greater  than  that  on  the  other. 
Find  the  two  rates  of  interest. 


Suggestion. 


24,000  X  j^  +  16,000  X  ^^  =840. 


13.  A  father  is  twice  as  old  as  his  son.  Twenty  years  ago  the 
father  was  six  times  as  old  as  his  son.     How  old  is  each  now  ? 

14.  If  the  length  of  a  rectangle  is  increased  by  3  feet  and 
its  width  decreased  by  1  foot,  its  area  is  increased  by  3  square 
feet.  If  the  length  is  increased  by  4  feet  and  the  width  de- 
creased by  2  feet,  the  area  is  decreased  by  3  square  feet. 
What  are  the  dimensions  of  the  rectangle  ? 

Note  that  if  w  and  I  are  the  original  width  and  length  of  the  rectangle, 
the  term  Iw  will  cancel  out  of  both  equations. 

15.  A  steamer  on  the  Mississippi  makes  6  miles  per  hour 
going  against  the  current  and  19^  miles  per  hour  going  with 
the  current.  What  is  the  rate  of  the  current  and  at  what  rate 
can  the  steamer  go  in  still  water  ? 

16.  One  number  is  three  times  another.  If  10  is  added  to 
each  of  the  numbers,  their  quotient  is  ^.     Find  the  numbers. 


222      SIMULTANEOUS   EQUATIONS   OF  THE   FIRST   DEGREE 


17.  A  starts  at  8  a.m.  for  a  walk  in  the  country.  At  10  a.m. 
B  starts  on  horseback  to  overtake  A,  which  he  does  at  noon. 
If  the  rate  of  B  had  been  one  mile  per  hour  less,  he  would 
have  overtaken  A  at  1  p.m.     At  what  rate  does  each  travel  ? 

18.  A  camping  party  sends  a  messenger  with  mail  to  the 
nearest  post  office  at  5  a.m.  At  8  a.m.  another  messenger  is 
sent  out  to  overtake  the  first,  which  he  does  in  2\  hours.  If 
the  second  messenger  travels  5  miles  per  hour  faster  than  the 
first,  what  is  the  rate  of  each  ? 

19.  There  are  two  numbers  such  that  3  times  the  greater  is 
18  times  their  difference,  and  4  times  the  smaller  is  4  less  than 
twice  the  sum  of  the  two.     What  are  the  numbers  ? 

20.  A  picture  is  3  inches  longer 
than  it  is  wide.  The  frame,  which  is 
4  inches  wide,  has  an  area  of  360 
square  inches.  What  are  the  di- 
mensions of  the  picture  ? 

y  =  X  +  S. 

2.4x  +  2.4y  +  4.42  =  360. 


s 

4" 

y^z  +  3 

X 

^y 

Suggestion. 


21.  The  difference  between 
two  sides  of  a  rectangular  wheat 
field  is  30  rods.  A  farmer  cuts 
a  strip  5  rods  wide  around  the 
field,  and  finds  the  area  of  this 
strip  to  be  1^  acres.  What  are 
the  dimensions  of  the  field? 


y=x-¥30 


Suggestion. 


t/  =  X  +  30. 


2  .  5(1/  -  10)  +  2  .  5(x-  10)+  4  .  52  =  7.^  •  160  =  1200. 

22.  In  a  number  consisting  of  two  digits,  the  sum  of  the 
digits  is  10.  If  the  order  of  the  digits  is  reversed,  the  num- 
ber is  decreased  by  54.     What  is  the  number? 


PROBLEMS   INVOLVING   TWO    UNKNOWNS  223 

23.  The  sum  of  the  length  and  width  of  a  certain  field  is 
260  rods.  If  20  rods  are  added  to  the  length  and  10  rods  to 
the  width,  the  area  will  be  increased  by  3800  square  rods. 
What  are  the  dimensions  of  the  field  ? 

24.  In  a  number  consisting  of  two  digits  the  sum  of  the 
digits  is  12.  If  the  order  of  the  digits  is  reversed,  the  number 
is  increased  by  36.     What  is  the  number  ? 

25.  A  bird  attempting  to  fly  against  the  wind  is  blown  back- 
ward at  the  rate  of  7^  miles  per  hour.  Flying  with  the  wind 
when  it  is  ^  as  strong,  the  bird  makes  48  miles  an  hour.  Find 
the  rate  of  the  wind  and  the  rate  at  which  the  bird  can  fly  in 
calm  weather. 

26.  There  is  a  number  whose  two  digits  differ  by  2.  If  the 
digit  in  units'  place  is  multiplied  by  3  and  the  digit  in  tens' 
place  is  multiplied  by  2,  the  number  is  increased  by  44.  Find 
the  number,  the  tens'  digit  being  the  larger. 

27.  In  a  number  consisting  of  two  digits  the  units'  digit  is 
equal  to  twice  their  difference.  If  the  order  of  the  digits  is  re- 
versed, the  number  is  increased  by  18.     Find  the  number. 

28.  If  the  length  of  a  rectangle  is  doubled  and  8  inches  added 
to  the  width,  the  area  of  the  resulting  rectangle  is  180  square 
inches  greater  than  twice  the  original  area.  •  If  the  length  and 
width  of  the  rectangle  differ  by  10,  what  are  its  dimensions  ? 

29.  There  is  a  number  consisting  of  three  digits,  those  in  tens' 
and  units'  places  being  the  same.  The  digit  in  hundreds'  place 
is  4  times  that  in  units'  place.  If  the  order  of  the  digits  is  re- 
versed, the  number  is  decreased  by  594.     What  is  the  number  ? 

30.  A  man  rowing  against  a  tidal  current  drifts  back  2^ 
miles  per  hour.  Rowing  with  this  current,  he  can  make  14^ 
miles  per  hour.  How  fast  does  he  row  in  still  water  and  how 
swift  is  the  current  ? 


224      SIMULTANEOUS   EQUATIONS   OF   THE   FIRST  DEGREE 

31.  Flying  against  a  wind  a  bird  makes  28  miles  per  hour, 
and  flying  with  a  wind  whose  velocity  is  2|  times  as  great,  the 
bird  makes  46  miles  per  hour.  What  is  the  velocity  of  the 
wind  and  at  what  rate  does  the  bird  fly  m  calm  weather  ? 

32.  A  freight  train  leaves  Chicago  for  St.  Paul  at  11  a.m. 
At  3  and  5  p.m.  respectively  of  the  same  day  two  passenger 
trains  leave  Chicago  over  the  same  road.  The  first  overtakes 
the  freight  at  7  p.m.  the  same  day,  and  the  other,  which  runs 
10  miles  per  hour  slower,  at  3  a.m.  the  next  day.  What  is 
the  speed  of  each  ? 

33.  Two  boys,  A  and  B,  trying  to  determine  their  respective 
weights,  find  that  they  balance  on  a  teeter  board  when  B  is  6 
feet  and  A  is  5  feet  from  the  fulcrum.  If  B  places  a  30-pound 
weight  on  the  board  beside  him,  they  balance  when  B  is  4  feet 
and  A  is  5  feet  from  the  fulcrum.     How  heavy  is  each  boy  ? 

34.  $  10,000  and  i$  8000  are  invested  at  different  rates  of 
interest,  yielding  together  an  annual  income  of  $  820.  If  the 
first  investment  were  $  12,000  and  the  second  $  6000,  the  yearly 
income  would  be  $  840.     Find  the  rates  of  interest. 

SIMULTANEOUS   EQUATIONS  IN   THREE  UNKNOWNS 

156.  An  equation  of  the  first  degree  in  jr, /,  and  z  is  one  which 
contains  these  letters  in  sucli  a  way  that  no  one  of  them  is 
multiplied  by  itself  or  by  any  other  one.  For  instance,  such  an 
equation  must  not  contain  such  terms  as  x"^,  y"^,  z^,  xy,  xz,  xyz,  etc. 

E.g.     2 .2;  -f  8  ?/  —  2  =  5  is  of  the  first  degree  in  x,  ?/,  and  z. 

157.  Systems  of  Equations.  Two  or  more  equations  involv- 
ing the  same  unknowns  form,  when  taken  together,  a  system  of 
equations.  A  system  of  three  equations  in  three  unknowns  may 
be  simultaneous  or  contradictory,  independent  or  dependent,  in 
the  same  sense  as  explained  in  §  147  for  two  equations  in  two 
unknowns. 


EQUATIONS  IN  THREE  UNKNOWNS        225 

Illustrative  Problem.  Three  men  were  discussing  their  ages 
and  found  that  the  sum  of  their  ages  was  90  years.  If  the  age 
of  the  first  were  doubled  and  that  of  the  second  trebled,  the 
aggregate  of  the  three  ages  would  then  be  170.  If  the  ages  of 
the  second  and  third  were  each  doubled,  the  sum  of  the  three 
would  be  160.     Find  the  age  of  each. 

Solution.     Let  x,  y,  and  z  represent  the  number  of  years  in  their  ages 

in  the  order  named. 

Then,  x  +  y  -\-  z  =90,  (1) 

2a:  +  32/  +  ^=  170,  (2) 

and  x-{-2y  -{-2z  =  160.  '  (3) 

If  we  subtract  equation  (1)  from  (2),  we  obtain  a  new  equation  from 

which  z  is  eliminated. 

That  is,  x  +  2y=S0.  (4) 

Again,  multiplying  equation  (2)  by  2  and  subtracting  (3), 

Sx-^4y  =  IS0.  (5) 

Equations  (4)  and  (5)  involve  the  two  unknowns  x  and  y.     Solving 

these  by  eliminating  y,  we  find        x  =  20.  (6) 

Substituting  x  =  20  in  (4),  y  =  30.  (7) 

Substituting  x  and  ?/  in  (1),  z  =  40.  (8) 

Check  by  showing  that  the  values  of  x,  ?/,  and  z  satisfy  the  original 

equations  and  the  conditions  of  the  problem. 

The  values  of  x,  y,  and  z  as  thus  found  constitute  the  solu- 
tion of  the  given  system  of  equations. 

Evidently  x  could  have  been  eliminated  first,  using  equations  (1) 
and  (2),  and  tlien  (1)  and  (3),  giving  a  new  set  of  two  equations  in  y 
and  z.     Let  the  student  find  the  solution  in  this  manner. 

Also  find  the  solution  by  first  eliminating  y,  using  (1)  and  (2),  and 
then  using  (2),  (3),  getting  two  equations  in  x  and  z,  from  which  the 
values  of  x  and  z  can  be  found. 

It  will  be  found  that  the  solutions  are  the  same,  no  matter  in 
what  order  the  equations  are  combined.  This  indicates  that 
G  system  of  three  independent  and  simultaneous  equations  of  the 
first  degree  in  three  unknowns  has  one  and  only  one  solution. 

As  in  the  case  of  two  equations,  each  should  be  first  reduced 
to  a  standard  form  in  which  all  the  terms  containing  a  given 
unknown  are  collected  and  united  and  all  fractions  removed. 


226      SIMULTANEOUS   EQUATIONS   OF  THE   FIRST   DEGREE 


EXERCISES 

Solve  the  following  systems,  and  check  the  results : 


2. 


3. 


8. 


9. 


10. 


\2x  —  y-{-z  =  lSy 
\x-2y-\-3z  =  10,  11. 

[Sx-\-y-4.z  =  20. 

5  X  —  3  y  +  z  =  W, 

x  —  Sy  —  z  =  —  3,  12. 

.2x—y-\-z  =  S. 

'4:X-\-2y  +  z  =  lS, 

■  X—  y  -\-  z=  A,  13. 
.  X  -\-2y  —  z  =  l. 

6a?H-4i/  —  42  =  —  4, 

■  Ax-2y-\-Sz  =  0,  14. 
.x-^y  +  z  =  4:. 

'x-{-2y  +  3z  =  5, 
4:X  —  3y  —  z=5,  15. 

.x-\-y  +  z  =  2. 

2x-8y-\-3z=2, 
x  —  4.y-{-5z  =  l,  16. 

\3x-10y-z  =  5. 

x  +  y-^z  =  l, 

x-h3y  +  2z  =  S,  17. 

[2x+Sy-3z  =  15. 

2  X  —  3  y  -\-  z  =  5j 
3x-\-2y-z  =  5j  18. 

x  +  y  i-z  =  3. 

x-{-y-\-z  =  6j 
I  3  X  -  2  2/  -  z  =  13,  19. 

[2x-y-\-3z  =  26. 

X  +  y  -{-  z  =  6, 

4  a?  —  ?/  —  2  =  —  1,  20. 

.2x-\-y  —  3z  =  —  Q. 


2x  —  3y  —  4^z  =  17, 

4  a; -4?/ -1-2  2  =  -10, 

7x-{-7y-{-5z  =  17. 

x-\-y-\-z  =  0, 

5  .T  -f-  3  2/  +  4  2;  =  —  1, 
[2x-7y  +  6z=21. 

\x  +  2y-z  =  2, 
2x  —  y-\-z  =  3, 
x-\-2y-\-z  —  S. 

2x—  y  —z  =  6, 
2x-2y  +  z  =  10, 
.x-\-y  —  3z  =  —  2. 

'  X  —  y  —  z  =  l, 
2x  +  3y  +  z  =  20, 
x  —  2y-\-z  =  0. 

Dx  —  2y-\-3z  =  5, 
—  2x  +  y-z  =  -ly 
.  —x  —  y  +  2z  =  -i. 

f4a-36-h2c  =  8, 
a-h?)  —  4c  =  —  16, 
7  a  —  46-l-c  =  4. 

5  ??i  —  4  71  +  ?'  =  8, 
3  ??i  +  n  —  3  r  =  0, 
t2?7i-4n  +  6r=28. 

9.1'  — 4?/  — 2;  =  — 4, 
2.u-|-5i/-6z  =  -12, 
-x4-2?/  +  42  =  30. 

4a.- -1-7?/  + 02  =  -3, 
a;  —  3 1/  -h  2  z  =  16, 
.5x4-2?/  —  42  =  1. 


PROBLEMS   INVOLVING  THREE   UNKNOWNS  227 

PROBLEMS   INVOLVING   THREE   UNKNOWNS 

158.  Illustrative  Problem.  A  broker  invested  a  total  of 
$  15,000  in  the  street  railway  bonds  of  three  cities,  the  first 
investment  yielding  3  %,  the  second  3|-  %,  and  the  third  4  %, 
thus  securing  an  income  of  $535  per  year.  If  the  second 
investment  was  one  half  the  sum  of  the  other  two,  what  was 
the  amount  of  each  ? 

Solution.     Suppose  x  dollars  were  invested  at  3  %,  y  dollars  at  3^  %, 

and  z  dollars  at  4  %. 

Then,                                r                     a;  +  2/ +  s  =  15000,  (1) 

.03a:+ .035?/  +  .04  2;  =  535,  (2) 

and                                        I                            X  -{-  z=^2y.  (3) 

From  equation  (3),                            x —  ly -\- z  =  ^.  (4) 

Subtracting  (4)  from  (1),                                ^y  =  15000.  (5) 

Hence,                                                                   y  =  5000.  (6) 

From  (1),  by  M,            .035  x  +  .035  y  +  .035  z  =  525.  (7> 

Subtracting  (7)  from  (2),       —  .005  x  +  .005  z  =  10.  (8) 

Dividing  (8)  by  .005,                               -x  +  z  =  2000.  (9) 

Substituting  (6)  in  (4),                               x  +  z  =  10000.  (10) 

Adding  (9)  and  (10),                                        2z=  12000.  (11) 

z  =  6000.  (12) 

Substituting  (6)  and  (12)  in  (1),                      x  =  4000.  (13) 
Hence,  $4000,  §5000,  and  $6000  were  the  sums  invested. 

WRITTEN   PROBLEMS 

Solve  the  following  problems,  using  three  unknowns  : 

1.  The  sum  of  three  angles.  A,  B,  and  C,  of  a  triangle  is 
180  degrees,  -i-  of  ^  +  ^  of  ^  +  4-  of  (7  is  48  degrees,  while  i 
of  A  -^  ^  of  -B  -|-  i  of  (7  is  30  degrees.  How  many  degrees  in 
each  angle  ? 

2.  The  combined  weight  of  1  cubic  foot  each  of  compact 
limestone,  granite,  and  marble  is  535  pounds.  1  cubic  foot  of 
limestone,  2  of  granite,  and  3  of  marble  weigh  together  1041 
pounds,  while  1  cubic  foot  of  limestone  and  1  of  granite  to- 
gether weigh  195  pounds  more  than  1  cubic  foot  of  marble. 
Find  the  weight  per  cubic  foot  of  each  kind  of  stone. 


228      SIMULTANEOUS  EQUATIONS   OF   THE   FIRST   DEGREE 

3.  A  number  is  composed  of  3  digits  whose  sum  is  7.  If 
the  digits  in  tens'  and  hundreds'  places  are  interchanged, 
the  number  is  increased  by  180 ;  and  if  the  order  of  the 
digits  is  reversed,  the  number  is  decreased  by  99.  What  is 
the  number  ? 

4.  The  sum  of  the  angles  A,  B,  and  C  of  a  triangle  is  180 
degrees.  If  B  is  subtracted  from  C,  the  remainder  is  -^  of  A, 
and  when  C  is  subtracted  from  twice  A,  the  remainder  is  4 
times  B.     How  many  degrees  in  each  angle  ? 

5.  The  sum  of  three  numbers,  a,  b,  and  c,  is  35.  Twice  a  is 
5  less  than  the  sum  of  b  and  c,  and  twice  c  is  4  more  than  the 
sum  of  a  and  b.     What  are  the  numbers  ? 

6.  If  X  is  the  number  of  seconds  in  the  Eastern  inter- 
collegiate record  for  a  mile  run,  y  the  number  in  the  Western 
record,  and  z  the  number  in  the  world's  record,  then 

x-\-y-^z  =  768.95, 
x-\-2  y  +  z  =  518.95, 
2x-y-\-z  =  502.75. 

7.  If  X  is  the  number  of  seconds  in  the  Eastern  inter- 
collegiate record  for  a  half  mile  run,  y  the  number  in  the 
Western  intercollegiate  record,  and  z  the  number  in  the 
world's  record,  then 

2x-^3  y-\-z=  692.9, 

Sx  +  2y-\-2z=  804.6, 

2x-y  -\-z  =  226.5. 

8.  If  X  is  the  number  of  seconds  in  the  world's  mile  trotting 
record  in  1806,  y  is  the  number  of  seconds  in  the  world's  record 
in  1885,  and  z  is  the  number  of  seconds  in  the  world's  record 
in  1911,  then  ,  x  +  y  +  z  =  A26.2o, 

2x-{-4:y  +  i^z  =  1584, 
-x  +  y-\-2z  =  lS6.75. 


REVIEW   QUESTIONS  229 

9.  Diophantus  of  Alexandria  (see  page  41)  gives  the  follow- 
ing problem  :  "  Find  three  numbers  such  that  the  sum  of  each 
pair  is  a  given  number." 

It  is  interesting  to  note  that  Diophantus  states  his  problem  in  words  in 
its  general  form,  but  he  solves  it  for  a  special  case  ;  viz.,  for  x  +  y  =  20j 
y  -\-  z  =  30,  z  +  X  =  40.  The  Greeks  did  not  use  letters  to  represent  num- 
bers in  general.     Hence  they  had  no  formulas  such  as  we  now  have. 

Solve  this  special  case. 

10.    Solve  the  preceding  problem  when  the  given  numbers 

are  a,  b,  c.     That   is,  solve  the   system  x  -{-y  =:a;  y  -\-z  =  b) 

and  z-\-x  =  G. 

REVIEW   QUESTIONS 

1.  Why  is  a  single  equation  in  two  unknowns  called  inde- 
terminate? 

2.  When  are  two"  such  equations  called  simultaneous  ?  AVhen 
independent  ?    When  contradictoi'y  9 

3.  May  two  equations  be  simultaneous  without  being  inde- 
pendent ?     May  they  be  independent  but  not  simultaneous  ? 

4.  Give  the  proper  description  to  each  of  the  following  sys- 
tems of  equations : 


x  +  y  =  \0      '  '     I  6  :c  +  3  ?/  =  21      '  '    i  2  .^•  +  4  ?/  =  10 

5.  Describe  elimination  by  the  process  of  addition  or  sub- 
traction; also  by  the  process  of  substitution.  Under  what  con- 
ditions is  one  or  the  other  of  these  methods  preferable  ? 

6.  Describe  the  solution  of  a  system  of  three  linear  equations 
in  three  unknowns.  Is  it  immaterial  which  of  the  three  varia- 
bles is  eliminated  first  ? 

7.  Can  you  find  a  definite  solution  for  two  equations  such 
as  4:X  —  S  y  —  z  =  o  and  x-\-y  -\-z  =  2? 

Eliminate  z  fl*om  these  equations.     W^hat  is  the  nature  of 
the  resulting  equation  ?     (See  §  147.) 


CHAPTER   XV 

GRAPHIC  REPRESENTATION* 

159.  Graphic  Representation  of  Statistics.  A  graphic  represen- 
tation of  the  temperatures  recorded  on  a  certain  day  is  shown 
on  the  next  page.     The  readings  were  as  follows : 

3  P.M.  29°  9  P.M.  21°  3  A.M.  12°  9  a.m.  12° 

4  P.M.  29°  10  p.m.  20°  4  a.m.  11°  10  a.m.  13° 

5  P.M.  28°  11  P.M.  17°  5  a.m.  10°  11  A.M.  16° 

6  P.M.  26°  12  m't.  16°  6  a.m.  10°  12  Noon  17° 

7  P.M.       24°  1  A.M.       14°  7  A.M.       10°  1  P.M.       18° 

8  P.M.     22°  2  A.M.     12°        8  A.M.     10°         2  p.m.     20° 

In  the  graph  each  heavy  dot  represents  the  temperature  at  a  certain 
hour.  The  distance  of  the  dot  from  the  heavy  vertical  hne  indicates  the 
hour  of  the  day  counted  from  noon,  and  its  distance  above  the  heavy  hori- 
zontal line  indicates  the  thermometer  reading  at  that  hour.  The  Hnes 
joining  these  dots  complete  the  picture  representing  the  gradual  changes  of 
temperature  from  hour  to  hour. 

Graphs  of  this  kind  are  used  in  commercial  houses  to  represent  varia- 
tions of  sales,  fluctuations  of  prices,  etc.  They  are  used  by  the  historian 
to  represent  changes  in  population,  fluctuations  in  the  amount  of  mineral 
productions,  etc.  In  algebra  they  are  used  in  solving  problems  and  in  ex- 
plaining many  difficult  processes.  In  the  succeeding  exercises  cross-ruled 
paper  is  essential. 

Make  a  graphic  representation  of  the  tables  of  data  on  the 
opposite  page : 

In  each  case  the  number  to  be  represented  by  one  space  on  the  cross- 
ruled  paper  should  be  chosen  so  as  to  make  the  graph  go  conveniently  on 
a  sheet.  Thus  in  Example  1  let  one  small  horizontal  space  represent  two 
years  and  one  vertical  space  a  million  of  population  ;  and  in  Example  3 
let  one  horizontal  space  represent  one  year  and  one  large  vertical  space 
one  hundred  thousand  of  population. 

*  Chapter  XV  may  be  omitted  without  destroying  the  continuity. 

230 


GRAPHIC   REPRESENTATION 


231 


" 

+ 

30 

J 

- 

s 

^ 

\ 

+ 

OR; 

p 

' 

V 

\ 

1 

V 

\ 

V 

ii 

20 

[ 

^ 

\ 

> 

\ 

1 

— 

0) 

\ 

/ 

s 

/ 

-H 

15 

-, 

V 

j 

— 

03 

> 

J 

V 

( 

\ 

/' 

s 

) 

-h 

ir» 

> 

s 

/ 

t 

o 

1 

, 

12 

M. 

bF 

.M 

6 

t 

12 

?A 

.M 

e 

) 

1 

2 

P.M. 

T 

imle  Lii'ie 

■■ 

^ 

1 

1 

I 

1 

1 

EXERCISES 

1.    The  population  of  the  United  States,  1800  to  1910 


1800  . 

.     4.3  (millions) 

18-tO  . 

.  17.1 

1880  . 

.  50.2 

1810  . 

.     7.2 

1850  . 

.  23.2 

1890  . 

.  62.6 

1820  . 

.     9.6 

1860  . 

.  31.4 

1900  . 

.  76.3 

1830  . 

.  12.9 

1870  . 

.  38.6 

1910  . 

.  92.0 

2.    The  population  of  the  boroughs  now  constituting  Greater 
New  York  City : 

1800  .  .     79  (thousands)     1840  .   .     391  1880  .  .  1912 


1810  . 

.  119 

1860  . 

.     696 

1890  . 

.  2507 

1820  . 

.  152 

1860  . 

.  1175 

1900  . 

.  3427 

1830  . 

.  242 

1870  . 

.  1478 

1910  . 

.  4767 

232 


GRAPHIC    REPRESENTATION 


160.  Axes.  In  tlie  graphs  thus  far  constructed  two  lines  at 
right  angles  to  each  other  have  been  used  as  reference  lines. 
These  lines  are  called  axes.  The  location  of  a  point  in  the 
plane  of  such  a  pair  of  axes  is  completely  described  by  giving 
its  distance  and  direction  from  each  of  the  axes. 


+^ 

q: 

(-1.3) 

■1-3 

CO 

Y'i 

(3,2) 

X 

< 

II 

;?s 

+1 

—1 

-3 

-^ 

-J 

t 

L 

V.l 

+3 

+4 

r: 

(-2.0.) 

' 

X 

-AXIS 

-J 

-2 

-i 

i 

: 

cJ 

H 

) 

k: 

(-8. 

r— 

\) 

-i 

^ 

_^ 

This  scheme  of  locating  points  by  two  reference  lines  is  already  familiar 
to  the  pupil  from  geography,  where  cities  are  located  by  latitude  and 
longitude  ;  that  is,  by  degrees  north  or  south  of  the  equator  and  east  or 
west  of  the  meridian  of  Greenwich. 

The  direction  to  the  right  of  the  vertical  axis  is  denoted  by 
a  positive  sign,  and  to  the  left,  by  a  negative  sign ;  while  di- 
rection upward  from  the  horizontal  axis  is  also  called  positive, 
and  downward,  negative. 


GRAPHIC    REPRESENTATION 


233 


The  horizontal  line  is  usually  called  the  jr-axis  and  the  ver- 
tical line  the^-axis. 

Abscissa.  The  perpendicular  distance  of  any  point  from 
the  ?/-axis  is  called  the  abscissa  of  the  point. 

Ordinate.  The  perpendicular  distance  of  any  point  from  the 
cc-axis  is  called  its  ordinate. 

Coordinates.  The  abscissa  and  ordinate  of  a  point  are  to- 
gether called  its  coordinates. 

E.g.  the  abscissa  of  point  P  in  the  opposite  figure  is  3  and  its  ordinate 
2,  or  we  may  say  the  coordinates  of  P  are  3  and  2,  and  indicate  it  thus, 
P:(3,  2),  writing  the  abscissa  first.  In  like  manner  f(ir  the  other  points 
we  write  (^  :  (  -  1,  3),  i?  :  (  -  2,  0),  >S' :  (  -  3,  -  4),  and  T :  (2,  -  3). 

We  see  that  in  this  manner  every 
point  in  the  plane,  corresponds  to  a 
pair  of  numbers,  and  that  every  pair 
of  numbers  corresponds  to  a  point. 

Quadrants.  The  two  axes  divide 
the  plane  into  four  parts  called 
quadrants.  These  are  numbered  I, 
II,  III,  IV,  in  counter-clockwise 
order,  as  shown  in  the  figure. 

Origin.  The  intersection  of  the 
two  axes  is  called  the  origin  of  coordinates. 

Examples.     1.    Plot  the  point  (4,  —  3). 

Solution.  Since  the  abscissa  is  positive,  we  measure  4  units  from  the 
?/-axis  to  the  Hght. 

Since  the  ordinate  is  negative,  we  measure  3  units  doxon  from  the  x- 
axis. 

Hence,  the  required  point  is  in  the  fourth  quadrant. 

The  student  should  use  squared  paper  and  plot  the  point. 

2.    Plot  the  point  (-2,  -  3). 

Solution.  Since  the  abscissa  is  negative,  we  measure  2  units  from  the 
2/-axis  to  the  left. 

Since  the  ordinate  is  negative,  we  measure  3  units  down  from  the  x-axis. 
Hence,  the  point  is  in  the  third  quadrant. 


-I 

I 

I 

U 

I 

:i 

r 

ir 

^_ 

L 

. 

234  GRAPHIC   REPRESENTATION 

EXERCISES 

1.  With  any  convenient  scale,  locate  the  following  points : 

(2,  6),  (  -  3,  5),  (0, 1),  (1,  0),  (0,  0),  (0,  -  1),  (0,  -  5),  (  -  5,  0), 
(2i  51),  (-  4,  -  8),  (3,  -  10),  (-  10,  3). 

2.  Locate  the  following  series  of  points  and  then  see  if  a 
straight  line  can  be  drawn  through  them  :  (0,  0),  (1,  1),  (2,  2), 
(3,  3),  (4,  4),  (  _  1,  -  1),  (  -  2,  -  2),  (-  3,  -  3). 

3.  Locate  the  following  and  connect  them  by  a  line :  (1,  0), 
(1,  2),  (1,  3),  (1,  4),  (1,  5),  (1,  -  2),  (1,  -  3),  (1,  -  4),  (1,  -  5). 
Name  other  points  in  this  line. 

4.  Draw  the  line  every  one  of  whose  points  has  its  hori- 
zontal distance  —  2 ;  also  draw  the  line  every  one  of  whose 
points  has  its  vertical  distance  -|-  3. 

5.  Locate  the  following  points  and  see  if  a  straight  line  can 
be  passed  through  them:  (1,  0),  (0,  1),  (2,  -1),  (3,  -2), 
(4,  -  3),  (  -  1,  +  2),  (  _  2,  3),  (  -  3,  4),  (  -  4,  5),  (i  i),  (i  |), 
(-|,  ^).     Can  you  name  other  points  on  this  line  ? 

Points  on  a  Straight  Line.  In  some  of  the  preceding  exer- 
cises, a  series  of  points  has  been  found  to  lie  on  a  straight  line, 
as  in  Examples  2,  3,  and  5.  Evidently  this  could  not  happen 
unless  the  points  were  located  according  to  some  definite 
scheme  or  law. 

In  Example  5  it  is  easy  to  see  what  the  law  is ;  namely,  the 
sum  of  the  abscissa  and  ordinate  is  1  for  each  point. 

Thus,  for  the  first  point  the  sum  is  1  +  0  =  1  ;  for  the  second  point  it 
is  0  +  1  =  1  ;  for  the  third,  2  +(—  1)=  1,  etc. 

For  the  last  point  the  sum  is  |  +  i  =  1,  for  the  next  to  the  last  it  is 
1  +  ^  =  1,  etc. 

Hence,  in  Example  5,  if  we  let  x  stand  for  any  one  of  the 
abscissas,  and  y  for  its  corresponding  ordinate,  we  have 
x-\-y  =  1  as  the  law  by  which  all  these  points  are  located. 

In  Example  2,  the  law  is  x  =  y,  and  in  Example  3  it  is  x  =  1, 
whatever  y  may  be.  In  Example  4,  the  laws  are  x  =  —  2  for 
the  first,  and  ?/  =  3  for  the  second. 


GRA]»HIC   REPRESENTATION  OF   EQUATIONS 


235 


GRAPHIC  REPRESENTATION  OF  EQUATIONS 

161.   Not  only  may  statistics  be  represented  by  graphs,  but 
there  is  also  a  way 
to   make  a  graphic 
representation  of  an 
equation. 

Example.  Make 
a  graph  of  the  equa- 
tion ic  -j-  ?/  =  3. 

Solution.  Writing 
the  equation  in  the  form 
y  =  3  —  cc,  we  see  that 
for  every  value  which 
we  give  to  x,  we  can 
find  a  corresponding 
value  of  y  so  as  to  satisfy  the  equation  y  =  Z  —  x. 
For  instance,  if  a:  =  0,  y  =  3  —  0  =  3  ; 

ifa;  =  2,  j/  =  3-2  =  l; 
if  X  =  3,  y  =  3  —  3  =  0,  etc. 
In  this  way  we  may  construct  the  following  table  of  values  of  x  and  y, 
and  we  may  extend  this  in  both  directions  as  far  as  we  like. 


.  lix  = 

-2 

-  1 

0 

1 

2 

3 

4 

5  etc., 

then  y  = 

5 

4 

3 

2 

1 

0 

-1 

-  2  etc. 

If  we  call  each  value  of  x  the  abscissa  of  a  point  and  the  corresponding 
value  of  y  its  ordinate,  this  table  gives  the  following  points  : 

A  B  c  D  E  F  G  n 

(-2,5)     (-1,4)    (0,3)     (1,2)     (2,1)     (3,0)     (4,-1)     (5,-2). 

If  we  plot  these  points,  as  in  the  figure,  we  see  that  they  all  lie  on  the 
straight  line  AH. 

Furthermore,  it'can  be  shown  that  all  the  points  whose  abscissas  and 
ordinates  are  values  of  x  and  y  which  satisfy  this  equation  lie  on  this 
straight  line  AH. 

Hence  the  line  AH  is  called  the  graph  of  the  equation  x  +  y  =  3. 


236 


GRAPHIC    REPRESENTATION 


162.  Graph  of  a  Linear  Equation.  If  an  equation  is  of  the 
first  degree  in  x  and  y,  all  the  points  whose  coordinates  are 
values  of  x  and  y  which  satisfy  this  equation  lie  on  a  straight 
line.     This  straight  line  is  called  the  graph  of  the  equation. 

An  equation  which  has  a  straight  line  for  its  graph  is  called 
a  linear  equation. 

To  graph  a  linear  equation  it  is  necessary  to  locate  only  two 
of  its  points  and  to  draw  a  straight  line  through  them.  If  this 
line  is  extended  indefinitely  in  both  directions,  it  will  pass 
through  all  the  points  whose  coordinates  satisfy  this  equation, 
and  through  no  other  points. 

E.g.  In  graphing  the  equation  x  —  y  =  5,  we  choose  a;  =  0  and  find 
y  =  —  5,  and  choose  y  =  0  and  find  x  =  5  and  plot  the  points  (0,  —  5)  and 
(5,  0).     The  fine  through  these  points  is  the  one  required. 

Example.     Graph  the  equation  3  ic  -}-  4  ?/  =  12. 

Solution.     We  see  that  when  x  =  0,  4i/  =  12,  and  y  =  3. 
Also  when  y  =  0,  3  x  =  12,  and  a;  =  4. 


^ 

~ 

n 

-t-r 

- 

" 

n 

" 

s 

S 

+r. 

k 

0 

N 

+  1-. 

S 

X) 

\ 

+ 

'i 

'S 

V 

^ 

k 

+!o 

V 

s, 

S 

+ 

\ 

-, 

-c 

-f 

hi 

l-f 

Y^ 

t^ 

!-• 

hf 

h- 

l-l 

> 

" 

u 

* 

Vg 

k 

s 

S 

s 

\ 

•^ 

k 

^ 

4 

V 

\ 

_ 

_ 

_ 

,K 

_ 

_ 

_ 

^ 

_ 

_ 

_ 

_ 

_ 

Hence  two  required  points  are  (0,  3)  and  (4,  0). 

Plotting  these  points,  we  draw  the  straight  line  through  them  as  shown 
in  the  figure. 

Check.  Plot  the  following  points,  whose  coordinates  satisfy  the  equa- 
tion, and  show  that  they  lie  on  the  graph. 

x  =  8  fx=12         \x  =  -'i     a;=-8 

y=-3       [y=-6       it/  =  6  y=9 


GRAPHIC  REPRESENTATKJN  OF  EQUATIONS 


237 


EXERCISES 

Construct  the  graph  for  each  of  the  following  equations : 

1.  3x-\-2y  =  1.         5.    5  —  2  ?/  =  6 X.  9.    3x  —  4y  =  —  7. 

2.  ox  —  Sy  =  —3.      6.    3a;+5?/=— 15.    10.    3x-4?/=— 12. 

3.  Tx-{-10y  =  2.        7.    2x-y=().  11.    7 y  =  9  x  -  63. 

4.  x-\-2y  =  0.  8.    3x  —  4:y=7.  12.    x  =  5y  -\-3. 

163.    Graphic  Solution.     Graph  on  the  same  axes  x  -\-  y  =  A 
and  y  —  X  =  2,  and  thus  solve  this  pair  of  equations. 


s 

1 

7 

' 

~^ 

"" 

s 

/ 

k 

+ 

/ 

s 

/ 

1 

\ 

/ 

s 

/ 

s 

/ 

s 

,+ 

4 

/ 

/ 

1 

s 

/ 

\ 

/ 

\ 

/ 

f 

o 

s 

I'l 

' 

>> 

/ 

K' 

■>> 

/ 

\ 

/ 

s 

/ 

s 

/ 

t' 

> 

\ 

^ 

s 

/ 

s 

/ 

s 

/ 

\ 

/ 

+ 

\ 

/ 

\ 

/ 

s 

/ 

S 

3 

— 

o 

/ 

- 

0 

+ 

t 

•2 

+ 

3 

S 

-1- 

4 

/ 

s 

^ 

^ 

/ 

\ 

/ 

■ 

\ 

/ 

\ 

^ 

_ 

_ 

u 

L 

_ 

- 

.1. 

k 

Solution.  The  two  graphs  are  found  to  intersect  in  the  point  (1,  3). 
Since  the  point  Hes  on  both  Hnes,  its  coordinates  should  satisfy  both 
equations,  as  indeed  they  do.  Since  these  lines  have  only  one  point  in 
common,  there  is  no  other  pair  of  numbers  which,  when  substituted  for 
the  unknowns  x  and  y,  can  satisfy  both  equations. 

Check.  Solve  these  equations  by  elimination,  and  see  if  you  get  these 
same  values  for  x  and  y. 

Hence,  x  =  l,  y  =  3  is  the  solution  of  this  pair  of  equations. 

Since  x  =  l,  y  =  3  represents  a  point  whose  abscissa  is  1  and 
whose  ordinate  is  3,  we  may  write  this  solution  as  we  would 
indicate  a  point;  namely,  (1,  3). 


238 


GRAPHIC   REPRESENTATION 


164.  Independent  and  Simultaneous  Equations.  Since  the 
graphs  of  the  preceding  equations  are  distinct,  the  equations 
are  properly  called  independent.  Since  the  graphs  intersect, 
the  coordinates  of  the  common  point  satisfy  both  equationc 
simultaneously. 

165.  The  Graph  shows  that  there  is  only  one  Solution.  Since 
two  straight  lines  intersect  in  but  one  point,  it  follows  that 
two  linear  equations  ivhich  are  independent  and  sirnultaneous  have 
one  and  only  one  solution.  Two  such  equations  may  be  solved 
by  finding  the  coordinates  of  the  point  ivhere  their  graphs  meet. 

EXERCISES 

Graph  the  following  and  thus  solve  each  pair  of  equations : 


|2.T-32/=:25, 


2. 


[  5  a;  -I-  G  2/  =  7, 


2.T-?/  =  -10. 


9. 


x-2y  =  2, 
2x-y=z-2. 

5x—Ty  =  21, 
x  —  4:y  =  —  l. 

5x-h2y  =  S, 
^2x-3y=:-12. 


6x-\-Sy  =  16, 
^2x-3y  =  ll. 

{ 3x  —  4:y  =  1, 

[2x  —  7y  =  o. 

ly-\-3x  =  7, 

\2y^X  =  -6. 

166.  Dependent  Equations.  If  we  attempt  to  plot  two  equa- 
tions which  are  not  independent,  such  as  x-\-y  =  1  and 
2x-\-2y  =  2,  the  graphs  will  be  found  to  coincide.  Such  equa- 
tions are  properly  called  dependent,  since  any  point  which  lies, 
on  the  graph  of  one  also  lies  on  the  graph  of  the  other. 

167.  Contradictory  Equations. 
If  we  attempt  to  plot  two  equa- 
tions which  are  not  simultaneous, 
such  as  X  -\-  y  =  1  and  x  -\-  y  =  2, 
the  graphs  will  be  found  to  be  par- 
allel and  hence  they  have  no  point 
in  common.  Such  equations  are 
properly  called  contradictory,  since 
there  is  no  point  which  can  lie  on 
both  graphs. 


S  1 

' 

s 

s 

s 

s 

s 

s 

s 

s 

\ 

\ 

V 

l. 

s 

s 

X 

,, 

s 

s' 

^ 

Sj 

L^V 

s; 

Sj>^ 

s 

S>' 

^ 

s 

\ 

Ss 

s 

s 

^ 

•v 

\ 

s 

s 

V 

s 

s 

k 

^ 

REVIEW   QUESTIONS  239 

HISTORICAL   NOTE 

Graphic  Representation  of  Equations.  The  representation  of  equations 
by  means  of  lines  is  due  to  Ren6  Descartes  (1596-1650) .  (See  next  page. ) 
By  the  time  of  Descartes  the  work  of  Vieta,  Harriot,  and  others  had 
given  to  algebra  the  modern  form,  thus  perfecting  it  as  a  general  in- 
strument of  research.  The  first  important  use  to  which  it  was  put  was 
the  application  to  geometry  made  by  Descartes.  This  must  be  regarded 
as  one  of  the  greatest  contributions  of  all  time  to  mathematics. 

Not  only  is  it  possible  to  graph  linear  equations  as  we  have  done  here, 
but  it  is  found  that  many  important  curves  of  different  kinds  are  graphs 
of  equations  of  higher  degrees.  The  iK)ints  where  two  such  curves  meet 
furnish  the  solutions  of  the  equations  of  which  they  are  the  graphs  ;  and 
conversely,  the  algebraic  solutions  of  such  equations  tell  the  points  where 
the  graphs  intersect.  This  enables  us  to  use  the  operations  of  algebra  in 
solving  a  large  range  of  problems  pertaining  to  lines  and  curves. 

The  graphic  method  is  much  used  by  engineers  and  others  in  the  solu- 
tion of  practical  problems. 

REVIEW    QUESTIONS 

1.  How  may  a  point  in  a  plane  be  located  by  reference  to 
two  fixed  lines  ?  What  are  these  lines  called  ?  What  names 
are  given  to  the  distances  from  the  point  to  the  fixed  lines  ? 

2.  Draw  a  pair  of  axes  in  a  plane  and  locate  the  following 
points:  (5,  0),  (-  2,  0),  (0,  3),  (0,  -  1),  (0,  0). 

3.  How  many  pairs  of  numbers  can  be  found  which  satisfy 
the  equation  x  —  2y  =  Q>?  State  five  such  pairs  and  plot  the 
corresponding  points.  How  are  these  points  situated  with 
respect  to  each  other  ?  What  can  you  say  of  all  points  corre- 
sponding to  pairs  of  numbers  which  satisfy  this  equation? 
What  is  meant  by  the  graph  of  an  equation  ? 

4.  How  many  pairs  of  numbers  will  simultaneously  satisfy 
the  two  equations  3  x  +  2  ?/  =  7  and  x  -\-  y  =3?  Show  by 
means  of  a  graph  that  your  answer  is  correct. 

5.  If  negative  numbers  could  not  be  used,  how  would  the 
graph  of  X  H-  2/  =  3  be  limited  ?     The  graph  of  a;  —  ^  =  3  ? 


CHAPTER   XVI 

SQUARE  ROOTS  AND  RADICALS 

168.    Square  Root.     A  square  root  of  a  number  is  one  of  its- 
two  equal  factors.     See  §  96. 

Thus  3  is  a  square  root  of  9,  since  3-3  =  9.  Similarly  a  +  b  is  a 
square  root  of  a^  +  2  a6  +  b^. 

It  should  be  noted  that  every  square  has  two  square  roots 
which  are  numerically  equal  with  opposite  signs. 

E.g.  —  3  is  a  square  root  of  9  as  well  as  +  3,  since  ( —  3)  •  (—  3)  =  9. 

Radical  Sign.  The  positive  square  root  of  a  number  is  in- 
dicated by  the  radical  sigri  V  ^alone  or  preceded  by  the  sign  -f . 
The  negative  square  root  is  indicated  by  the  radical  sign  pre- 
ceded by  the  sign  — . 

E.g.    +  \/9  or  \/9  =  +  3  and  not  —  3,  and  —  \/9  =  —  3,  and  not  +  3. 

The  square  root  of  any  number  is  at  once  evident  if  we  can 
resolve  it  into  two  equal  groups  of  factors. 

V576  =  V2.2-2.2.2.2.3.3  =  \/(23  .  3)(28  .  3)  =  V'24T24  =  24. 

ORAL  EXERCISES 

Find  the  following  indicated  square  roots : 


1. 

V4. 

6. 

-V49. 

11. 

V196. 

16. 

-V025. 

2. 

V9. 

7. 

V81. 

12. 

-V256. 

17. 

-  V900. 

3. 

-Vl6. 

8. 

VlL^l. 

13. 

-  V144. 

18. 

Vioooo. 

4. 

V25. 

9. 

~Vl()9. 

14. 

V'400. 

19. 

-V64. 

6 

V36. 

10. 

V225. 

15. 

V289. 

20. 

-  V1600. 

240 


Rene  Descartes  (1596-1650)  was  born  near  Tours  in  France, 
and  died  in  Stockholm.  On  leaving  school  he  went  to  Paris 
and  gave  two  years  to  the  study  of  mathematics.  After  spending 
some  time  in  the  army  and  in  travel  he  finally  settled  in  Paris 
and  devoted  himself  to  philosophy  and  mathematics. 

In  the  year  1629  Descartes  moved  to  Holland,  in  order  to  pursue 
his  studies  without  interruption.  From  that  time  he  lived  in  seclu- 
sion, carrying  on  his  correspondence  with  learned  men  through 
one  or  two  trusted  friends  who  kept  his  exact  whereabouts  a 
profound  secret.  In  1649  he  was  invited  to  Stockholm  by  Queen 
Christiana  of  Sweden,  and  here  he  died  the  following  year. 

Descartes  may  be  regarded  as  the  first  of  the  modern  mathe- 
maticians. 


SQUARE    ROOTS   AND    RADICALS  241 

Dividing  Exponents  by  2.     We  see  that  the  square  root  of  an 
even  pov:er  is  obtained  by  dividing  its  exponent  by  2. 
Thus,  we  have  Va*  =  a",  which  is  a^^^. 
Similarly,  Va^  =  a^  which  is  a^^. 

And  Vci^  =  a^  which  is  a^^. 

ORAL   EXERCISES 

Find  the  following  indicated  square  roots : 


1. 

V2^. 

11. 

V3^. 

2. 

V512. 

12- 

^/5««^ 

3. 

-V78. 

13. 

vr^-' 

4. 

Vai2. 

14. 

Vc'-- 

5. 

-V3". 

15. 

VaTi^^. 

6. 

Va24. 

16. 

—  Va^". 

7 

-V3^. 

17. 

Vz/^. 

6. 

V^. 

18. 

-  Va;8^ 

9. 

Va^". 

19. 

Vmi6". 

10. 

V42». 

20. 

Vp°'. 

21, 

V(a  +  b)\ 

22 

V(a  +  6)l 

2   . 

-V(tt  +  6)2". 

24. 

V(a  -  6)'^. 

25. 

-  V(a  —  ?>)■•^ 

26. 

V(a  -  6)1^ 

27. 

-V(a2-6)^ 

28. 

-V(a-52)2-. 

29. 

V(a-/>  +  c/. 

30.    V(a  —  6  +  c)«^ 

169.  The  Square  Root  of  a  Product.  The  square  root  of  the 
product  of  several  factors,  each  of  which  is  a  square,  may  be 
found  by  taking  the  square  root  of  each  factor  separately,  as  in 
the  following  examples : 


(1)  V4  .  16  .  25  =  V4  .  Vl6  .  V2o  =  2  •  4  •  5  =  40. 

This  is  true  since  4  •  16  •  25  can  be  written  as  the  product  of  two  groups 
of  equal  factors  (2  •  4  •  5)(2  •  4  .  5).  Hence,  one  of  these  (2  •  4  •  5)  is  the 
square  root  of  4  •  16  •  25. 

(2)  V25a^  =  V52 .  Vo^  .  V6"6  =  5  a''b\ 

This  is  true  since  b'^a^h^  can  be  written  as  the  product  of  two  groups  of 
equal  factors  (5  aP'h^){b  a-b^).  Hence,  one  of  these  (5  a'^b^')  is  the  square 
root  of  5-  a'^b^. 

From  these  examples  we  see  that  the  square  root  of  a  product 
may  he  obtained  by  dividing  the  exponent  of  each  factor  by  2. 


242  SQUARE   ROOTS   AND   RADICALS 

ORAL   EXERCISES 

Find  the  following  indicated  square  roots  : 

6.    V25T36.  11.    -V¥c^, 


1. 

-  V22 .  32. 

2. 

V81  .  121. 

3. 

V49  •  25  .  169. 

4. 

-V82.  52.32. 

5. 

V54 .  3^- .  4^ 

7.  -  V312 .  5".  12.    V32a;2?/2. 

8.  -  V222  .  312.  13.    V9  xY\ 


9.    Vl6a262c2.  14.    -Vl21a2a;^ 


10.    V64aV.  15.    -VT^a^ft*. 

Notice  that  V9  4  16  is  not  equal  to  V9  +  VI6. 
The  preceding  exercises  illustrate 

Pr^'Qcipl^  XVIII 

170.  Rule.  T/ie  sq^'jare  root  of  a  product  is  obtained  by 
finding  the  square  root  of  each  factor  separately  and 
then  taking  the  product  of  these  roots.    That  is, 


Va  •  b  =  Va  •  V6. 

HISTORICAL  NOTE 

Radicals.  — The  essential  elements  in  the  theory  of  radicals  were  de- 
veloped long  before  the  present  notation  came  into  use.  Alkarismi,  the 
author  of  the  first  Arabian  algebra  (see  page  32),  states  in  substance  that 
av'6  =  Va^,  ^Ja  •  Vb  =  Vab  (Principle  XVIII). 

Nicolas  Chuquet,  in  a  French  book  published  in  1484,  gives  the  earliest 
known  use  of  the  radical  sign,  though  the  Hindus  had  used  a  similar 
symbol. 

Christoff  Rudolff,  a  German  writer,  used  the  radical  sign  in  1525  but  he 

indicated  cube  root  by  V  V  V    and  fourth  root  by  V  V  . 

Wallis  (see  page  101)  first  used  fractional  exponents  to  indicate  roots. 


Find  the  following 

ORAL  EXERCISES 

square  roots: 

1.    —  V4  a^b\ 

5.    _V64a26^ 

2.    -V3^x'Y\ 

6.    --^/l()Ul'b\ 

3.    V52.322P. 

7.    V5^m". 

4.    Vl21a^y2 

8.    V5^.38.7«. 

3.    VS^*-7^^a*. 


10.    -V25a26«ci2. 


11.  VsTxYc^. 

12.  V49a«yi"?^ 


SQUARE   ROOT   OF   A   POLYNOMIAL  243 

THE   SQUARE   ROOT  OF   A  POLYNOMIAL 

171.  Relations  between  a  Square  and  its  Square  Root.  If  we 
square  a-\-b  we  get  a^  +  2  ab  +  b-.  To  liud  the  square  root  of 
a^  -\- 2  ab  -\- b^  we  try  to  see  how  a  -\-b  can  be  derived  from 
a2  +  2  a6  +  b-. 

Step  (1)  We  see  that  a  can  be  found  from  a-  by  taking  its 
square  root. 

Step  (2)  We  see  that  b  can  be  found  from  2ab  by  dividing 
it  by  twice  the  term  already  found,  a. 

The  steps  in  the  work  are  arranged  as  follows : 

The  given  square  =  a^  -{-  2  ab  +  b- \a  +  h  =  the  square  root. 

The  square  of  a     =  a^ 

Trial  divisor  =  2  a  2  ab  +  b^  =  first  remainder. 

Complete  divisor  =2a  +  b  2  ah  -{-  b^  =  b(2  a  +  b). 

0 

Explanation.  Having  found  a,  as  indicated  in  step  (1)  above,  we  sub- 
tract its  square  from  a^  +  2  ab  +  b~. 

The  remainder  begins  with  2  ab,  which  we  use  to  find  b,  as  in  step  (2) 
above. 

That  is,  we  multipl}'-  a  by  2,  and  divide  2  ab  by  this  product,  2  a,  which 
we  call  the  trial  divisor,  thus  getting  6,  the  second  term  of  the  root. 

Now,  the  remainder,  2  ab  +  b'^,  may  be  written  (2  a  +  b)b.  Hence,  if 
we  add  b  to  2  a  (calling  2  a  +  b  the  complete  divisor) ,  and  multiply  this  by 
6,  the  product,  2  a&  +  6^,  is  the  rest  of  the  square. 

Thus,  in  finding  the  terms,  a  and  b,  of  the  root,  we  really  build  up  the 
whole  of  the  square  and  subtract  it  piece  by  piece.  The  remainder,  zero, 
indicates  that  the  square  root  is  exact. 

Example.     Find  the  square  root  of  S6x^  —  84  xy  +  49/. 

Given  square        =  36  x-  —  Si  xy  +  49  y^  [  6x—  7  y  i=  the  root. 

Square  of  6  a;        =  36  a;^ 

Trial  divisor  =  2  •  6  x         —  Sixy  -\- 49  y^. 

Complete  divisor  =12x  —  7y  —  Mxy  +  49  y-  =  —  7  y(12x  —  7  y). 

0 

Explanation.  The  first  term  of  the  root  is  6  x«ince  V36a-2  =  6x.  The 
product  of  twice  6  x  and  the  second  term  of  the  root  is  —  84  xy.  Hence, 
the  second  term  of  the  root  is  —  84  a-.v  -^  12  a-  =  —  7  x.  The  complete 
divisor  is  then  12  a;  —  7  y,  and  the  rest  of  the  square  is  —  7  y{12  x  —  7  y) 
=  -Sixy  +  49  y^. 


244  SQUARE   ROOTS   AND    RADICALS 

EXERCISES 

Find  the  square  root  of  each  of  the  following : 

1.  16  a:^  -  64  a^2/' 4- 64  2/^.  6.    121  -  44  ir^+ 4  a^. 

2.  64ar^-32a^  +  4.  7.   9  a' -  30  ab' -{- 25  b\ 

3.  25  +  49ar^-70a;.  8.   81  a- -216  a  +  144. 

4.  a^'  +  ea'b  +  db^  9.    4  a^fe^  -  44  a^^  +  121. 

5.  36a^-84aj  +  49.  10.    81  -  270  ?/ +  225  2/1 

172.  Rule  for  Finding  the  Square  Root  of  a  Polynomial.  From 
the  squares  ^^  _^  ^y  =  «2  ^  2  a6  +  b^ 

(a  +  6  +  cf  =  [(a  +  6)  +  c]^  =  (a  +  bf  +  2(a  +  6)c  +  c^, 
(a  +  b  +  c  ^  ciy  =  l(a  +  b  +  c)  +  d] 

=  (a  4-  6  +  c)2  +  2(a  +  ^>  +  c)d  +  d%  etc., 
we  see  that 

(1)  If  to  a"  we  add  2  ab -hb^  =  (2  a-\-  b)b,  we  get  (a  +  6)2 ; 

(2)  If  to  (a  +  6)'  we  add  2(a  +  6)c  +  c' 

=  [2 (a  +  6)  +  cy,  we  get  (a  +  6  +  c)^ ; 

(3)  If  to  (a  +  ?;  +  c)2  we  add  2(a  +  6  +  c)d  +  d^ 

=  [2(a  +  6  +  c)  +  d]  d,  we  get  (a  +  6  +  c  +  df,  etc. 

Hence,  in  squaring  a  polynomial : 

i^o?"  ei>ert/  neiv  term  added  to  the  root  there  is  a  new  part  added 
to  the  power.  This  new  part  consists  of  twice  the  sum  of  the  pre- 
ceding terms  of  the  root  plus  the  last  term  of  the  root,  all  midtiplied 
by  the  last  term  of  the  root. 

This  is  expressed  by  the  formula : 

(a  +  6  +  c  +  flO' 

=  a^  +  (2  a  +  6)6  +  [2(a  +  b)  +  c']c  +  [2(a  +  6  +  (?)  +  (/](/. 

WRITTEN  EXERCISES 

Write  the  following^  squares  in  the  above  form : 

1.  {x-\-y  +  z)\  4.    {2a  +  b-\-?>c-\-df. 

2.  {x  +  y-\-z-\-vy.  5.    {b  +  Ac  +  2d  +  ey. 

3.  (a;  +  2?/  +  3z  +  4v)2.  6.    {x  +  2a +  3b  +  cf. 


SQUARE   ROOT   OF   A   POLYNOMIAL  246 


Example  1.     Find  the  square  root  of 

9  X*  -12  a^  -{-2S  x"  -16  X  -{- 16. 

Solution.  a    -\-   b  -h  c 

Square  root    =  3  x'^  —  2  a;  +  4 


Given  square  =  9  x*  -  12  x^ -\- 2S  x^  -  IQ  x -}- W 

a-  =  (3  a;2)2  =  9^4 

2  rt  =  2  .  3  ic2  =  6  x'^  -  12  a:3  +  28  x2  -  16  X  +  16 

(2  a  +  &)6  =  (6  x2  -  2  x)(-  2  X)  =  -12x3+    4x^ 

2(a  +  6)  =  6x2-4x  24x2-16x+16 

[2(a -f  ft)+c]c  =  (6x--4x  +  4).4=  24  x^  -  16  x  +  16 

0 
Explanation.     The  first  term  of  the  root  is  Vdlc^  =  Sx^.     This  is  sub- 
tracted from  the  square. 

The  second  term  of  the  root  is  —  12  x^  ^  (2  •  3  x^)  =  —  2  x. 
The  second  part  of  tlie  square  is  (6  x^  —  2  x)  (—  2  x)  =  —  12  x-^  +  4  x^, 
corresponding  to  (2  a  +  b)b.    This  is  now  subtracted. 
The  third  term  of  the  root  is  24  x^  -^  (2  •  3  x^)  =  4. 
The  third  part  of  the  square  is  (6  x'^  —  4  x  +  4)  •  4  =  24  x^  —  16  x  +  16, 
corresponding  to  [2(a  +  6)+  c'\c.     This  is  now  subtracted. 

Thus  at  each  step  a  new  term  of  the  root  is  found  by  dividing  the  first 
term  of  the  remainder  by  twice  the  first  term  of  the  root ;  and  then  a 
new  part  of  the  power  is  built  up  and  subtracted. 

Since  the  final  remainder  is  zero,  the  square  root  is  exact. 

Example  2.     Find  the  square  root  of 

16  x^  -  24  x''  +25  x"  -  52  a^3  _,_  34  ^2  _  20  x  +  25. 

Solution. 

a         +6      +  c  +  d 
Square  root  4  x^  —  3  x'^  +  2  x  —  5 


Given  square         1 6  x*^  —  24  x^  +  25  x*  -  52  x^  +  34  x^  -  20  x  +  25 

a-  =  (4  x3)2  =  10^ 

-  24  x5  +  25  x^  -  52  x^  -(-  34  x^  -  20  x  +  25 

(2a-hb)b=  -24x5+    9x* 

16  x*  -  52  x-i  +  34  x2  -  20  X  +  25 

(2(a  +  &)+c)c=  16x^-12x-^+    4x2 

-  40  x3  +  30  x2  —  20  X  +  25 
[•2(a  +  b  +  c)+d^d=  -  40  x^  +  30  x^  -  20  x  +  25 

Explanation.     In  this  solution  only  the  successive  parts  of  the  formula 
are  written  down.     Let  the  student  give  the  explanation  in  full. 


246  SQUARE   ROOTS  AND   RADICALS 

From  the  preceding  examples  we  have  the  following 

Rule :  (1)  Arj^ange  the  polynomial  according  to  as- 
cending or  descending  powers  of  some  letter. 

(2)  Find  tlie  square  root  of  the  first  term,  and  write  it 
as  the  first  term  of  the  root. 

(3)  Subtract  the  square  of  this  first  term  of  the  root. 

(4)  Divide  the  first  term  of  tlze  remainder  by  twice  the 
first  term  of  the  root,  and  write  the  quotient  as  the  second 
term  of  the  root. 

(5)  Add  this  second  term  of  the  root  to  twice  the  first 
term,  and  multiply  the  sum  by  tlve  second  term.  This 
product  is  the  second  part  of  the  square  and  is  to  be  sub- 
tracted. 

(6)  Kow  use  the  sum  of  the  first  two  terms  of  tize  root  to 
find  the  third  term,  just  as  the  first  term  was  used  to  find 
the  second ;  and  continue  in  this  manner  till  all  the  terms 
of  the  root  are  found. 

EXERCISES 

Find  the  square  root  of  each  of  the  following  : 

2.  l-2a  +  3«2-2a3  +  a*.  5.    c''-4c3  +  6c2-4c  +  l. 

3.  l+2  6-?>2_2  63+54^  6.    a^-2ar'  +  5a:2_4^_^4_ 

7.  a^  +  4  a?h  +  6  o?h^  +  4  a^^  +  h\ 

8.  .t"  —  4  a.-^^  +  6  x'^'if  —  4  xy^  +  y^. 

9.  a"  +  53  a^  +  14  a^  +  28  a  +  4. 

10.  a'^-f 6a^-f-15a^  +  20a3+15a2+6a  +  l. 

11.  a*^  -  6  a^  +  15  a^  -  20  a^  +  15  a-  -  G  a  -|-  1. 

12.  4  x^  -  12  r*  +  13a;4  _  14  x^  4-  13  .r-  4  x  -|-4. 

13.  IG  a«  +  24  a''  +  25  a' +20  a'  +  lOa^  -|-  4  a  +  1. 

14.  x'^y^  +  2  xy  +  3  xY  H-  4  x^f  +  3  .i-y  -}-  2  xy  +  1 . 

15.  1  4-  2  X  +  3  x2  -h  4  ar^  4-  5  a;^  +  4  .r^  4-  3  x'«  +  2  x'  +  ^- 


SQUARE   ROOT   OF  AN   ARITHMETIC   NUMBER        247 

THE   SQUARE   ROOT   OF   AN    ARITHMETIC   NUMBER 

173.  Rule  for  Finding  the  First  Term  of  the  Square  Root  of  an 
Integral  Number.  Since  V  =  l  and  9^  =  81,  the  square  of  a 
number  of  one  figure  contains  either  one  or  two  figures. 

Since  10^=  100  and  99^  =  9801,  the  square  of  a  number  of 
two  figures  contains  either  three  ov  four  figures. 

Similarly,  the  square  of  a  number  of  three  figures  contains 
either  j^ve  or  six  figures,  and  so  on. 

Rule.     Hence,  to  find  th-e  first  figure  in  the  root 

(1)  Separate  the  number  into  groups  of  two  figures  each, 
counting  from  units'  place  toward  the  left.  The  last  group 
may  contain  only  one  figure. 

(2)  Tahe  the  square  root  of  the  largest  square  in  the  left 
hand  group.  This  is  the  first  figure  of  tlxe  root,  and  there 
are  as  many  figures  in  the  root  as  there  are  groups  in  the 

number. 

Examples.  1.  To  find  the  first  figure  and  the  number  of  figures  in  the 
square  root  of  450,769  we  write  it  thus  45  07  69. 

Since  there  are  three  groups  of  two  figures  each,  the  square  root  con- 
tains three  figures,  and  hence  it  starts  with  the  hundreds'  figure. 

Since  36  is  the  largest  square  in  45,  the  first  figure  in  the  root  is 
V36  =  6. 

2.  Similarly,  the  square  root  of  6,762,436,  written  6  76  24  36,  con- 
tains four  figures  of  which  the  first  one  is  thousands'  figure. 

Since  4  is  the  largest  square  in  6,  the  first  figure  of  the  root  is  Vi  =  2. 

ORAL  EXERCISES 

Give  the  first  figure  in  the  square  root  of  each  of  the  following, 
and  state  whether  it  stands  in  units',  tens'  or  hundreds'  place : 

1.  8947.  5.    90,401.  9.    7347.  13.    107. 

2.  6205.  6.    63,401.  10.    73,470.  14.    4091. 

3.  19,140.  7.    1428.  11.    14,051.  15.    10,007. 

4.  72,048.  8.    194,670.         12.    140,051.         16.    100,007. 

The  square  root  of  an  arithmetic  number  may  be  found  by 
the  process  just  used  for  polynomials,  if  we  remember  that  a 
number  like  637  is  reall}^  a  polynomial,  namely  600  +  30  +  7- 


2-1:8  SQUARE   ROOTS   AND   RADICALS 

Illustrative  Example  1.     Find  the  square  root  of  405769. 

Solution.  Sqttaee  Square  Root 

a  +  b    +  c 
40  67  69    1 600  +  30+7  =  637 
a2  =  600-^  =  36  00  00 

2  a  =  1200  4  67  69 

b  =      30 


2a  +  6  =  1230 3  69  00  =(2a+6)6 

2(a+  &)=  1260  88  69 

c=        7 


2(a  + 6)+c  =  1267 88  69  =[2(a  +  &)+c]c 

0 

Explanation.  The  first  figure  in  the  root  is  the  square  root  of  the 
largest  square  in  the  left  hand  group,  and  since  there  are  three  groups, 
the  root  starts  with  600,  which  corresponds  to  a  of  the  formula  (§  172). 

Subtracting  the  square  of  600  we  have  4  67  69. 

The  trial  divisor  is  2  o  =  1200  and  when  4  67  69  is  divided  by  1200, 
the  largest  number  of  tens  in  the  quotient  is  3.  Hence  30  corresponds  to 
b  of  the  formula. 

The  complete  divisor  is  2  a  +  6  =  1230,  and  this  multiplied  by  b  gives 
(2  a  +  h)b  =  36900,  \vhich  is  the  second  part  of  the  square.  Subtracting 
36900,  the  remainder  is  88  69. 

The  next  trial  divisor  is  2(«  -\-b)=  1260  and  8869  -4-  1260  gives  7  as 
the  largest  number  of  units.     This  is  c  of  the  formula.     Then 

2(a  +  &)  +  c=  1267, 
and  this  multiplied  by  7  gives  [2 (a  +  &)  +  c]c  =  88  69. 

The  remainder  is  now  zero,  and  hence  the  square  root  is 

600  +  30  +  7  =637. 

In  case  a  square  consists  of  a  whole  number  and  a  decimal  part 
the  figures  in  the  integral  j^ciTt  of  the  square  root  are  found 
exactly  as  in  Example  1  above.  To  find  the  decimal  part  of 
the  root,  we  proceed  as  in  the  next  illustrative  example. 

ORAL  EXERCISES 

Give  the  first  figure  in  the  square  root  of  each  of  the  follow- 
ing, and  tell  in  which  place  it  stands; 

1.  12.645.  4.   941.61.  7.   49.29. 

2.  1.2645.  5.    94.16-  8.    4.929. 

3.  126.45.  6.    9.416  9.    492.9. 


SQUARE   ROOT   OF  AN   ARITHMETIC   NUMBER        249 

Illustrative  Example  2.     Find  the  square  root  of  67.7329. 
Solution. 


Square 

Square  Root 

67.73  29 

8  +  -2  +  .03  =  8.23 

a^=    82  = 

64 

2a  =  16 

3.73  29 

6=      .2 

2a  +  6=16.2 

3.24 

=  (2a  +  h)h 

2(a  +  &)=16.4 

.49  29 

c  =      .03 

.49  29 

2(a  +  6)+c  =  16.43 

=  [2(a  +  6)+c]c 

0 

Explanation.  There  is  only  one  group  of  two  figures  to  the  left  of  the 
decimal  point.  Hence  the  first  figure  of  the  root  is  units'  figure.  Since 
the  square  of  a  decimal  contains  twice  as  many  decimal  places  as  the 
numjber  itself,  there  will  be  one  decimal  figure  in  the  root  for  every  two 
in  the  square. 

In  getting  the  second  figure  of  the  root,  the  trial  divisor  is  2  a  =  16. 
The  quotient  is  6  =  .2  since  .2  x  16  =  3.2.  The  quotient  could  not  be  .3 
since  .3  x  16  =4.8. 

Similarly,  in  getting  the  third  figure,  we  divide  .4929  by  16.4  and  the 
quotient  is  .03  since  .03  x  16.4  =  .492. 

Illustrative  Example  3.     Find  the  square  root  of  9.1204 
Solution. 


Square 

Square  Root 

a  +  h 

9.1204 

3  +  .02  =  3.02 

a2  =  32  = 

9 

2a  =  6 

.12  04 

6=    .02 

a -1-6  =  6.02 

.12  04 

=  (2a  +  h)h 

0 

Explanation.  Since  there  is  only  one  group  to  the  left  of  the  decimal 
point,  the  first  figure  of  the  root  is  in  units'  place. 

In  this  case  in  dividing  ,1204  by  6,  the  quotient  is  .02  since  .02  x  6  = 
12  ;  that  is,  there  is  a  zero  in  tenths'  place,  and  there  are  only  two  terms 
to  the  root. 


250  SQUARE   ROOTS   AND   RADICALS 

EXERCISES 

Find  the  square  root  of  each  of  the  following : 

1.  294,849.         5.   3481.  9.   100,489.  13.  357.21. 

2.  37,636.  6.   7569.  10.   265.69.  14.  16,641. 

3.  872,356.  7.    1849.  11.    87.4225.  15.  32,761. 

4.  599,076.         8.   73,441.         12.    170,569.  16.  2332.89 

174.  The  First  Digit  of  the  Square  Root  of  a  Decimal  Number. 
In  case  a  number  has  no  integral  part,  the  first  term  of  its 
square  root  is  found  as  in  the  following  examples : 

1.  In  .1742  the  first  digit  in  the  root  is  .4  since  the  square 
of  .4  is  .16,  the  largest  square  in  .17. 

2.  In  .0542  the  first  digit  in  the  root  is  .2  since  the  square  of 
.2  is  .04,  the  largest  square  in  .05. 

3.  In  .0070  the  first  digit  in  the  root  is  .08  since  the  square 
of  .08  is  .0064,  the  largest  square  in  .0070. 

4.  In  .0007  the  first  digit  in  the  root  is  .02  since  the  square 
of  .02  is  .0004,  the  largest  square  in  .0007. 

From  these  examples  we  have  the  following 

Rule.  To  find  the  first  digit  in  the  square  root  of  a  decimal 
number  : 

(1)  Divide  the  munber  into  groups  of  two  fi.gures  each 
counting  from  the  decimal  point  toward  tJve  right,  adding 
a  zero  if  necessary  to  complete  the  last  group. 

(2)  Take  the  square  root  of  tlve  largest  square  contained 
in  the  first  group  which  is  not  all  zeros,  and  prefix  to  it  as 
many  zeros  as  there  are  complete  groups  of  zeros  to  the 
right  of  the  decimal  point. 

For  instance,  in  the  above  Examples  1  and  2,  there  are  no  groups  of 
zeros  to  the  right  (3f  the  decimal  point.  Hence  the  first  digit  in  the  root 
in  each  case  is  tenths'  digit. 

In  Examples  .3  and  4,  there  is  one  whole  group  of  zeros  to  the  right  of 
the  decimal  point.     Hence  the  first  digit  of  the  root  is  hundredths'  digit. 

Similarly,  when  there  are  two  complete  groups  of  zeros  to  the  right 
of  the  decimal  point,  the  first  digit  in  the  root  is  thousandths'  digit. 


SQUARE    ROOT   OF    AN    ARITHMETIC   NUMBER         251 

Illustrative  Example.     Find  the  square  root  of  .06783. 
Solution. 


SyUARK 

SyuAKE  Root 
a  +  &  +   c 

.06  78  30 

.2  +  .06  +  .0004 

0^  = 

.2'^ 

: 

=  .04 

2a  = 

2  X 

.2  =  .4 

.02  78 

b  = 

.06 

2a  +  b  = 

.46 

.02  76 

=  (2  a +  6)6 

2(a+  b)  = 

.52 

.00  02  30  00 

c  = 

.0004 

2(a  +  b)+c  = 

.5204 

.00  02  08  16 
.00  02  21  84 

=  [2(a  +  6)  + 

Explanation.  According  to  the  rule,  .2  is  the  first  term  of  the  root 
because  4  is  the  largest  square  in  6  and  there  is  no  group  preceding  .06.  The 
process  is  the  same  as  in  tlie  case  of  an  integral  square,  but  special  care 
is  now  needed  in  handling  the  decimal  points,  which  is  done  exactly  as 
in  operations  upon  decimals  in  the  process  of  division  in  arithmetic. 

For  instance,  in  finding  the  third  term  in  this  example,  we  divide 
.00023  by  2(.26)  =  .52  and  the  quotient  lies  between  .0004  and  .0005. 
Hence  c  =  .0004.  Zeros  are  annexed  to  .00023  to  correspond  to  the  number 
of  decimal  places  in  the  product  .5204  x  .0004. 

The  three  terms  of  the  root  thus  found  are  .2  +  .06  +  .0004  =  .2604. 

To  find  the  next  term  of  the  root  we  would  divide  .00002184  by 
2 (.2604)  =  ..5208,  finding  the  quotient  .00004.  We  would  then  add  .00004 
to  .5208  and  nmltiply  the  sum  by  .00004,  annexing  zeros  to  the  dividend 
as  before. 

175.  Approximate  Square  Roots.  Evidently  the  process  in 
this  example  may  be  carried  on  indefinitely.  .2604  is  an 
approximation  to  the  square  root  of  .06783 ;  in  fact,  the  square 
of  .2604  differs  from  .06783  by  only  .00002184.  The  nearest 
approximation  using  three  decimal  places  is  .260.  If  the 
fourth  figure  were  5,  or  any  digit  greater  than  5,  then  .261 
would  be  the  nearest  approximation  using  three  decimal  places. 
Hence,  four  places  must  be  found  in  order  to  be  sure  of  the 
nearest  approximation  to  three  places  ;  and  five  places  must  be 
found  in  order  to  be  sure  of  the  nearest  approximation  to  four 
places,  and  so  on. 


252  SQUARE   ROOTS   AND   RADICALS 

EXERCISES 

Find  the  square  root  of  each  of  the  following,  correct  to  two 
decimal  places : 

1.  387.  5.  51.  9.  5.  13.  .02. 

2.  5276.  6.  3.824.  10.  7.  14.  .003. 

3.  2.92.  7.  2.  11.  8.  15.  .5. 

4.  27.29.  8.  3.  12.  11.  16.  .005. 

SIMPLIFYING   SQUARE  ROOTS 

176.  Approximate  Square  Root  of  a  Whole  Number.  When 
we  wish  to  approximate  the  square  root  of  a  number  such  as  8 
we  make  use  of  Principle  XVIII  as  follows : 

V8  =  V4T2  =  V4T  V2  =  2  V2.  We  then  find  the  square 
root  of  2  and  multiply  by  2. 

Similarly  VT2  =  \/4  .  V3  =  2  v'3, 

V20=  Vi-  V5=:2V5;     \/32  =  Vie.  \/2  =  4V2; 
y/cfi  =  Va2  .  Va  =  ay/ a  ;  Va^  =  Va^  •  Va  =  a'^Va  ;  Va'^y^b  =  ay'^Vb. 

In  general  if  the  expression  under  the  radical  sign  contains  a 
factor  which  is  a  square  this  factor  may  he  removed  by  writing 
its  square  root  before  the  radical  sign. 

In  this  manner,  the  square  roots  of  a  few  small  numbers  like 
2,  3,  5,  etc.,  are  made  to  do  service  in  finding  the  roots  of  many 
large  numbers. 

ORAL  EXERCISES 

Change  the  following  so  as  to  leave  no  factor  which  is  a 
square  under  the  radical  sign. 


1. 

V8. 

5. 

V20- 

9. 

V27. 

13. 

V54. 

2. 

V12. 

6. 

V24. 

10. 

V50. 

14. 

Val 

3. 

V40. 

7. 

V28. 

11. 

V72. 

15. 

Va^ 

4. 

V18. 

8. 

V32. 

12. 

V45. 

16. 

Va^ 

SIMPLIFYING   SQUARE   ROOTS 


253 


17.  Va^. 

18.  Va". 

19.  Vo". 

20.  V^. 

21.  Vo^d. 

22.  Vo^. 


23.  Vo^. 

24.  Vo^ft. 

25.  Vo^. 

26.  Va^. 

27.  Va26"^ 

28.  Vo^^ 


29.  Vo^ftl 

30.  Vo^^e. 


35.    Vo^^. 


31.    -^/aWx. 


32.    Va^ 


x. 


36.  Va2"6*"a;. 

37.  Va^. 

38.  ^cFb^. 


33.    Va^ft^a;.         39.    VSa^^z, 


34.    Va^&V.        40.    Vl8aV>. 


177.  Approximate  Square  Root  of  a  Fraction.  A  fraction  is 
squared  by  squaring  its  numerator  and  its  denominator  separately, 

Cf  CL         CL 

since  -  x  -  =  — .     Hence,  to  extract  the  square  root  of  a  fraction, 
b      b      b^ 

we  find  the   square    root   of  its   numerator  and    denominator 

separately. 

^■9-     ^li  =  h  since  ^  x  |  =  ^f . 

When  we  wish  to  approximate  the  square  root  of  a  fraction, 
such  as  -|,  we  make  use  of  Principle  XVIII  as  follows,  in  order 
to  get  an  integer  instead  of  a  fraction  under  the  radical  sign : 

V|  =  VJ|  =  V^  VTo  =  i  VTo. 

In  each  case  after  multiplying  both  terms  of  the  fraction  by  the 
denominator,  the  fraction  is  resolved  into  two  factors,  one  of 
ichich  is  a  perfect  square. 

In  this  way  instead  of  getting  the  square  root  of  both  nu- 
merator and  denominator  of  such  a  fraction  as  f,  we  get  the 
square  root  of  10  and  divide  the  result  by  5. 

In  general, 

\/-  =  \/— =  \/—  •  a  =  \  —  '  Va  =  -  Va; 
^a      ^a'-      >'a2  V«2  ^ 

^'a      >'a2      \a2  \  a}  a 


254  SQUARE   ROOTS  AND   RADICALS 

178.   Rationalizing  the  Denominator.     Changing  a  radical  ex- 
pression so  as  to  leave  no  denominator  under  a  radical  sign  is 

called  rationalizing  the  denominator ;  thus,  \/-  =  -Va. 

^a     a 


4 
4 


2  V5  V3  \a 


3 


4 


4. 


ORAL 

EXERCISES 

iominat( 

3r  in  each  of 

4 

-Vi 

4 

-aI 

4 

-■  4e 

4 

14.    J 

4 

15.^? 

17. 


18. 


1^ 


4     '4     "-4      -W't' 
4     . 


179.  In  rationalizing  a  denominator  we  should  multiply 
the  terms  of  the  fraction  by  the  smallest  number  ivhich  icill 
make  the  denommator  a  perfect  square. 

Thus,  J±  =  Ja  =  Jl  .  Va  =  1  v^. 

Similarly,  a/I^ = J^  =  JIH  •  ^/;;f;z  =  J-  v;^ 

Again,  V|  =  V^  =  V J^  =  VS  .  V2  =  i  V2. 

If  the  fraction  is  of  the  form  — -,  we  multiply  both  nii- 

merator  and  denominator  by  V^;  thus 
a  a^b         a^b 


V6      y'b  '  Vb         b 

JL.g. := = • 

V3      V3  v/3        3 


since  V6-  V6  =  V6^  =  b. 


SIMPLIFYING   SQUARE   ROOTS  255 

ORAL  EXERCISES 

In  the  maimer  just  indicated  rationalize  the  denominators 
in  the  following : 


1-     \/--  4.    A/-.  7.    \l^    .  10. 

2. 


ir  1  1  'v 


^S■       -  xi  • 


3.    A/-,-  6.    a/^    .  9.    JX-  12.    J4* 

a^  ^'S  ^  ab  yi  ab 

180.    Simplified  Form  of  Radicals.     An  expression  in  one  of 

the  forms   Vo^^  ^Jv   ^f  ^^  ^^^^  ^^  ^^  simpUjied  when  it  is 

reduced  (1)  so  that  no  radical  occurs  in  a  denominator,  and 
(2)  so  that  no  factor  which  is  a  perfect  square  remains  under 
the  radical  sign. 

Such  radical  expressions  may  always  be  simplified  by  Prin- 
ciple XVIII. 

WRITTEN  EXERCISES 

Given  V2  =  1.411,  V3  =  1.732,  V^=  2.236,  compute  each 
result  in  Examples  1  to  15,  correct  to  two  places  of  decimals, 
without  further  extraction  of  roots  : 


1. 

V80. 

6. 

V2.3. 

11. 

V27  +  Vi 

2. 

Vi. 

7. 

V72. 

12. 

V45+Vi. 

3. 

Vi. 

8. 

V98. 

13. 

V5(:)-vj+V8. 

4. 

V48. 

9. 

V363. 

14. 

VJs  +  Vr^-Vs. 

5. 

V75. 

10. 

Vl2o. 

15. 

V32+V72-V18. 

Simplify  the  following: 

16.  V32  o?h.             19.    V45  Q(?ifb\  22.  v'500  x'a'b. 

17.  V81  x%\           20.    V63  bc>d\  23.  ^^x"  +  Qxy -\-3y\ 

18.  VSOoW.         21.    V900  ab'c\  24.  ^'^x^-12y\ 


256  SQUARE   ROOTS   AND   RADICALS 

EQUATIONS   SOLVED   BY  SQUARE    ROOTS 

181.  Since  2^  =  4  and  also  (-2)2  =  4,  it  follows  that  the 
equation  x'^=4:  has  two  roots,  namely  ic=2  and  x=  —2.  These 
are  usually  written  x=  ±2. 

This  solution  is  obtained  by  taking  the  square  root  of  both 
sides,  that  is,  by  dividing  both  sides  by  the  same  number. 

This  operation  may  now  be  added  to  those  enumerated  in 
Principle  VI  for  the  solution  of  equations. 

ORAL  EXERCISES 

Find  all  roots  of  the  following  equations  : 

1.  a;2  =  9.  11.  x'^  =  9a\  21.  .^2=  64  am^ 

2.  a;2  =  16.  12.  x''  =  Sa\  22.  x^  =  36rh\ 

3.  x2  =  25.  13.  x'^  =  16a\  23.  .^^  ^  81  s^^-^. 

4.  x''  =  3Q.  14.  x'^  =  ^9b\  24.  .1-2  =  50. 

5.  it;2  =  49.  15.  a;2  =  25a2.  25.  .^•2  =  72. 

6.  x'  =  64.  16.  x'  =  81  a262.  26.  .^2  =  98. 

7.  a;2^81.  17.  x^  =  A9a*b\  27.  a;2  =  32. 

8.  0.-2  =  100.  18.  ;v2  =  9  cc^b.  28.  ^2  =  49  a^ 

9.  a;2  =  8.  19.  .t'2  =  25(a  +  6)2.  29.  x''  =  S6a^b\ 
10.  a.'2  =  12.  20.  .i'2  =  50(a  -  by.  30.  x''  =  200  a\ 

WRITTEN  EXERCISES 

1.  Find  approximately  to  two  decimal  places  the  sides  of 
a  square  whose  area  is  120. 

2.  Approximate  to  two  decimals  the  side  of  a  square  having 
an  area  equal  to  that  of  a  rectangle  whose  sides  are  15  and  20. 

3.  How  many  rods  of  fence  are  required  to  fence  a  square 
piece  of  land  containing  50  acres,  each  acre  containing  160 
square  rods  ? 

4.  A  square  checkerboard  has  an  area  of  324  square  inches. 
What  are  its  dimensions? 


APPLICATIONS   OF   SQUARE   ROOT  257 

APPLICATIONS    OF   SQUARE  ROOT 

182.  The  Theorem  of  Pythagoras.  Some  of  the  most  interest- 
ing and  useful  applications  of  the  square  root  process  are  con- 
cerned with  the  sides  and  areas  of  triangles. 

The  fact  that  the  sum  of  the  squares  on  the  two  sides  of  a 
right  triangle  equals  the  square  on  the  hypotenuse  was  used 
in  Chapter  VIII.     See  page  151. 

If  a  and  h  are  the  lengths  of  the  sides, 

and  c  the  length  of  the  hypotenuse,  all 

measured  in  the  same  unit,  the  theorem 

of  Pythagoras  says : 

c'i  =  ai  +  b-\  (1) 

Hence,  by  .S',  cfi  =  c^  -  62,  (2) 

and  ?/2  ^  (.2  -  a2.  (3) 

Taking  the  square  root  of  both  sides  in  each  equation,  we  have 

c  =  Va2  4-  hK  (4) 

az=^c^-  h'K  (5) 

h  =  Vr2  -  a\  (6) 

The  negative  root  is  omitted,  since  the  side  of  a  triangle 
cannot  be  a  negative  number.  By  these  formulas,  if  any  two 
sides  of  a  right  triangle  are  given,  the  other  may  be  found. 

E.g.  if  a  =  4,  &  =  3,  then,  by  Eqnation  (4)  above, 

c  =  \/42  +  82  =:  Vl6  -I-  9  =  V25  =  5. 
Again  if  c  =  5,  6  =  3,  then,  by  Equation  (5)  above, 

a  =  V52  -  32  =  V25  -  9  =  Vf«  =  4  ; 
and  if  c  =  5,  a  =  4,  then,  by  Equation  (6), 

h  =  V52  -  42  =  \/25  -  16  =  V9  =  3. 

Example  1.  If  the  two  sides  of  a  right  triangle  are  8  and 
12,  tind  the  hypotenuse  correct  to  two  decimal  places. 


Solution.     We  have        c  =  Va^  +  b-  =  \/64  -I-  144  =  v/208, 

V2O8  =  VI6  .  13  =  VI6  .  Vl3  =  4  Vl3  =  4(3.605)  =  14.420. 

Example  2.     If  the  hypotenuse  of  a  right  triangle  is  10  and 
one  side  is  8,  find  the  other  side. 

Solution.     We  have  b  =  y/d^  -  d^  =  VlOO  —  64  =  \/36  =  6. 


258  SQUARE   ROOTS   AND   RADICALS 

PROBLEMS 

In  solving  the  following  problems,  simplify  each  expression 
under  the  radical  sign  before  extracting  the  root.  Find  all 
results  correct  to  two  decimal  places. 

1.  The  sides  about  the  right  angle  of  a  right  triangle  are 
each  15  inches.     Find  the  hypotenuse, 

2.  The  hypotenuse  of  a  right  triangle  is  9  inches  and  one 
of  the  sides  is  6  inches.     Find  the  other  side. 

Hint.    If  X  =  the  length  of  the  required  side,  then  06^  =  0^— 1)^=^1— Z^. 

3.  The  hypotenuse  of  a  right  triangle  is  25  feet  and  one  of 
the  other  sides  is  15  feet.     Find  the  other  side. 

4.  The  hypotenuse  of  a  right  triangle  is  12  inches  and  the 
other  two  sides  are  equal.     Find  their  length. 

Solution.     Let  s  be  the  length  of  one  of  the  equal  sides. 
Then,  s^  +  s^  =  114. 

2  s"  =  141. 
s2  =  72, 
s  =  V72  =  6  V2  =  6  X  1.414  =  8.484, 

5.  The  hypotenuse  of  a  right  triangle  is  30  feet  and  the 
other  sides  are  equal.     Find  their  length. 

6.  The  hypotenuse  of  a  right  triangle  is  c  and  the  sides 
are  equal.  Find  their  length.  Solve  Examples  4  and  5  by 
means  of  the  formula  here  obtained, 

7.  The  diagonal  of  a  square  is  8  feet.     Find  its  area. 

8.  The  side  of  an  equilateral  triangle  is  6  inches.  Find 
the  altitude, 

A  line  drawn  from  a  vertex  of  an  equilateral  / 

trianj^le  perpendicular  to  the  base  meets  the  base  at         y 
its   middle  point.      Hence   this   problem   becomes  :         / 
The  hypotenuse  of  a  rij^ht  triangle  is  G  and  one  side      / 
is  3.     Find  the  remaining:  side.  /       ^ 


APPLICATIONS   OF  SQUARE   ROOT  259 

9.  The  side  of  an  equilateral  triangle  is  10.  Find  the 
altitude. 

10.  The  side  of  an  equilateral  triangle  is  s.  Find  the 
altitude. 

This  is  equivalent  to  finding  a  side  of  a  right  triangle  wliose  hypote- 
nuse  is  s,  the  other  side  being  -  •     Let  h  equal  the  altitude. 


Then  h  =  yjs^  -  h^V  =  ^^s-^  - 


=a/*-^=v¥=^i 


>'4  2  Z 

Hence  h  =  -  \  3. 

2 

This  formula  gives  the  altitude  of  any  equilateral  triangle  in 
terms  of  its  side ;  namely,  the  altitude  of  an  equilateral  triangle 
is  equal  to  one  half  of  the  side  multiplied  by  V3. 

By  means  of  this  formula  solve  Examples  8  and  9. 

11.  Find  the  altitude  of  an  equilateral  triangle  whose  side 
is  4^.     Substitute  in  the  formula  under  Example  10. 

12.  Find  the  area  of  an  equilateral  triangle  whose  side  is  5. 

Since  the  area  of  a  triangle  is  ^  the  product  of  the  base  and  altitude, 
we  first  find  the  altitude  by  means  of  the  formula  under  Example  10,  and 
then  multiply  by  |  the  base. 

13.  Find  the  area  of  the  equilateral  triangle  whose  side  is  s. 

Show  the  result  to  be  -  V3. 

4 
Solution 

From  Example  10,  the  altitude  is 

^  =  -  V3. 
2 

The  area  of  a  triangle  is  equal  to  one  half  the  product  of  the  base  and 

the  altitude. 

Hence  Area  =  -  •  ?  \/3  =  ^  Vs. 

2     2  4 

That  is,  the  area  of  an  equilateral  triangle  is  equal  to  one  fourth  of  the 

square  of  its  side  multiplied  by  VS. 


260  SQUARE    ROOTS   AND   RADICALS 

14.  If   the   area   of   an    equilateral    triangle   is    16    square 
inches,  find  the  length  of  the  side. 

Let  s  equal  the  length  of  the  side.     Then  by  the  formula  derived  in 

s-    /- 
Example  13,  we  have  16  =  —  v  3. 

Hence,    ^^  ::.  -^  =  ^  V3  =  2L33  x  L732. 
V3      3 

15.  The  area  of  an  equilateral  triangle  is  50  square  inches. 
Find  its  side  and  altitude. 

16.  The  area  of  an  equilateral  triangle  is  a  square  inches. 
Find  the  side. 

Solve    the    equation   a  =  -  v  3    for   s,    and   simplify    the    expression, 

4 


finding    s^^  =  ^,^nds=^|^^^  =  ^■^/SaV3. 
V3  ^33 

17.    The  area  of  an  equilateral  triangle  is  240  square  inches. 

Find  its  side.     (Substitute  in  the  formula  ob- 

/\        7\  tained  in  Example  16.) 

V   \  V  \ 

/   ■     Y    7   \  18.    Find   the    area   of    a-  regular   hexagon 

\    t/Ix;    /       whose  side  is  7. 

^ — t — ^  A  regular  hexagon  is  composed  of  6  equal  equilateral 

triangles,  whose  sides  are  each  equal  to  the  side  of  the 

hexagon  (see  figure).     Hence  this  problem  may  be  solved  by  finding  the 

area  of  an  equilateral  triangle  whose  side  is  7,  and  multiplying  the  result 

by  6. 

19.  Find  the  area  of  a  regular  hexagon  whose  side  is  s. 
Solve  Example  18  by  substituting  in  the  formula  obtained  here. 

20.  The  area  of  a  regular  hexagon  is  108  square  inches. 
Find  its  side. 

H  the  area  of  the  hexagon  is  108  S(j[uare  inches,  the  area  of  one  of  the 
equilateral  triangles  is  18  square  inches. 

21.  The  area  of  a  regular  hexagon  is  a  square  inches.  Find 
its  side.  Solve  Example  20  by  substituting  in  the  formula  ob- 
tained here.  ^Ins.     5  =  4"\/2  a  V3. 


APPLICATIONS  OF   SQUARE   ROOT  261 

22.    Find  the  radius  of  a  circle  whose  area  is  9  square  inches. 

The  area  of  a  circle  is  found  by  squaring  the  radius  and  multiplying 
by  3.141(3.  The  number  3.1410  is  approximately  the  quotient  obtained 
by  dividing  the  length  of  the  circumference  by  the 
diameter  of  the  circle.  This  quotient  is  represented 
by  the  Greek  letter  ir  (pronounced  pi).  In  this  chap- 
ter we  use  3f  as  an  approximation  to  tt.  This  differs 
from  the  real  value  of  tt  by  less  than  .0013,  and 
hence  is  accurate  enough  for  most  purposes.  If  a 
represents  the  area  of  a  circle,  the  above  rule  may  be 

written 

a  =  irr-. 

Hence,  if  a  =  9,  r2  =  -=  —  =  —  =  2.863, 

'  TT      34      22 


and  r  =  ^72.803. 

23.    Find  the  radius  of  a  circle  Avhose  area  is  68  square  feet. 

REVIEW    QUESTIONS 

1.  What  is  meant  by  square  root  ? 

2.  State  in  words  the  principle 

Va  •  6  =  Va  •  V6. 

3.  How  is  this  principle  used  to  simplify  radicals  ?  Show  by 
use  of  this  principle  how  to  find  V28,  having  given  V7  =  2.696. 

/-  /-I 

4.  Why  is  ^  V3  considered  simpler  than  V^  or  — =  ?     Show 

how  the  value  of  the  following  may  be  approximated  by  find- 
ing only  one  square  root. 

5  V20  H-  2  V45  -  3  V80  -f  2  VJ. 

5.  Write  the  square  of  a  -j-  b  -\-  c  +  d  in  such  a  form  as  to 
derive  from  it  the  rule  for  finding  the  square  root  of  a  poly- 
nomial. 

6.  In  finding  the  square  root  of  an  arithmetic  number, 
how  is  it  divided  into  groups  (1)  in  case  of  a  whole  number, 
(2)  in  case  of  a  decimal  ? 


CHAPTER   XVII 

FURTHER   OPERATIONS  ON  RADICALS 

183.    Higher  Roots.     By  means  of  an  index  figure  the  radical 

sign  is  made  to  indicate  other  roots  than  square  roots. 

Thus,  the  cube  root  of  8,  or  one  of  its  three  equal  factors,  is  written 
VS  =  2.     The  fou7-th  root  of  16  is  written  VlG  =  2. 

Radical  Expressions.  Any  expression  which  contains  an  in- 
dicated root  is  called  a  radical  expression. 

The  expression  under  the  radical  sign  is  called  the  radicand. 

Rational  Numbers.  Integers  and  fractions  whose  terms  are 
integers  are  called  rational  numbers. 

E.g.    2  +  v^is  a  radical  expression.     5,  |,  ^-^^^  are  rational  numbers. 

A  surd  is  an  indicated  root  of  a  rational  number,  which  is 
not  reducible  to  a  rational  number. 

E.g.  \/2  is  a  surd  since  it  cannot  be  reduced  to  a  rational  number. 
Vi,  \/3  are  surds  for  the  same  reason.     Vl)  is  not  a  surd  since  V9  =  3. 

V  2  +  v'2  is  not  a  sui'd  since  2  +  \/2  is  not  a  rational  number. 

Order  of  a  Surd.     The  order  of  a  surd  is  indicated  by  the  index 

of  the  root.  • 

E.g.  \/4:  is  a  surd  of  the  third  order,  or  of  index  three  ;  y/3  is  a  surd  of 
the  fifth  order  or  of  index  five. 

Quadratic  Surd.  Surd  expressions  containing  no  indicated 
roots  except  square  roots  are  called  quadratic  surds. 

E.g.    V7,  V'l  4-  V3,    3  +  \/5,  — ^r- 1  are  quadratic  surda. 

\/7  -  v'5 

Mixed  Surd.  Entire  Surd.  The  product  of  a  surd  and  a 
rational  factor  is  called  a  mixed  surd.  A  surd  which  has  no 
rational  factor  is  called  an  entire  surd. 

E.g.    V2  is  an  entire  surd  ;  3V2  is  a  mixed  surd. 

262 


REDUCTION   OF   SURDS  263 

REDUCTION  OF   SURDS 

184.  By  reduction  of  a  surd  we  mean  the  changing  of  its  form 
ivithout  changing  its  value. 

185.  Reduction  of  a  Surd  to  its  Simplest  Form,  This  kind  of 
reduction  was  used  throughout  tlie  preceding  chapter  and  is 
based  on  Principle  XVIII ;  namely ;   Va  •  b  =  Va  •  V6. 

E.g.  V80=VT6T5=\/16.  V5  =4V5, 

and  V|  =  \/|  =  VP3=V|.  V3  =  ^v/3. 

186.  Reduction  of  a  Mixed  Surd  to  an  Entire  Surd.  It  is  some- 
times desired  to  place  the  rational  coeihcient  of  a  mixed  surd 
under  the  radical  sign.  This  is  called  reducing  a  mioced  surd  to 
an  entire  surd.     This  is  also  based  on  Principle  XVIII. 

Example.     Reduce  3V2  to  an  entire  surd. 
Since  3  =  \/9  we  have  S\2  =  \  9  •  V^=  V9T2  =  Vl8. 
In  this  form,  Principle  XVIII  may  be  stated  thus : 
TTie  product  of  the  square  roots  of  two  numbers  is  equal  to  the 
sqxiare  root  of  the  product  of  the  numbers. 

ORAL  EXERCISES 

Reduce  each  of  the  following  to  entire  surds. 

1.  2V2.  10.    aV6.  '  19.    (a-6)Va^. 

2.  2V5.  11.    ^V^.  20.  (a-b)yj~^^ 

3.  3V3.  12.    c-\  2.  "~^ 
/in                10     /.,   ,   r.\^/-             21.  3.r 


a  —  b 


v^- 


4.  3V10.  13.    {a  +  b)^c. 

5.  a^b.  14.    (a  —  6)Vc.  /~T 


22.   (x-^i^)yj- 


6.  aVab.  15.  (a-\-b)Va-b.                    ^    ^-^'  +  1' 

__ IT 

7.  xVxy.  16.  aVa-\-b.  23.  '^2^'-- 

8.  abVab.  17.  aVa  —  b.                              pr 

'  r  ..    / 24.   a^H/— . 

xyWx.  18.  {a-\-b)^a-^b.                 ^ab 


9 


264  FURTHER   OPERATIONS   ON   RADICALS 

187.    Simplifying  a  Surd  of  any  Order.     Reductions  of  surds  of 
any  order  may  be  made  by  means  of  the  following : 

(1)  The  nth  root  of  the  product  of  tivo  numbers  is  equal  to  the 
product  of  the  nth  roots  of  the  numbers. 

(2)  The  product  of  the  nth  roots  of  two  numbers  is  equal  to  the 
nth  root  of  the  j^roduct  of  the  numbers.     In  symbols : 

(1)   V^TTb^^a^Vl',         (2)   V~a.Vb  =  V^b. 

Example    1.     Simplify    ■\/x'^y^. 

^x'^y^  =  Vx^y^  •  xy'^  =  vx^y^  •  y/xy'^  =  xy  y/xy^. 
Example    2.     Reduce  2Va^  to  an- entire  surd. 
Since  2  may  be  written  V2^,  we  have 


Example  3.     Simplify  ^r— 


3 

3 


x^      3  9  art      3  L^3  -'/;^.3      3  y.  3 

3        \  27        >'27  N'27  3 


188.  These  examples  all  involve  the  principle  that  the  nth 
root  of  the  nth  power  of  ayiy  number  is  the  number  itself. 

In  symbols,  -y'^  —  q 

189.  Steps  in  Simplifying  a  Surd.  In  simplifying  a  surd 
of  any  order,  the  first  step  is  to  factor  the  radicand  so  that  one 
factor  shall  be  a  perfect  power  of  the  same  degree  as  the  root  indi- 
cated. 

Thus,  in  an  indicated  cube  root  we  find  the  greatest  factor 
which  is  a  perfect  cube  and  write  its  cube  root  before  the  radi- 
cal sign,  leaving  the  other  factor  under  the  radical  sign. 

E.g.  v/TG=v^8  .  \/2  =2v/2. 

In  an  indicated  fourth  root  we  find  the  greatest  factor  which 
is  a  perfect  fourth  power  and  write  its  fourth  root  before  the 
radical  sign. 

E.g.  v^32  =  v^  .  y/2  =  2  v/2. 


REDUCTION   OF   SURDS 


265 


ORAL  EXERCISES 

Simplify  the  following : 


1. 

^16. 

6. 

■\/x^y*. 

11. 

a/32. 

16. 

^a^b\ 

2. 

■^'32. 

7. 

^ab\ 

12. 

a/64. 

17. 

-voc^y^. 

3. 

^40. 

8. 

^32. 

13. 

^w>b. 

18. 

Vx^hj. 

4. 

<m. 

9. 

a/48. 

14. 

-\Ui*b\ 

19. 

Vx^'y. 

5. 

Vx^y^. 

10. 

^^64. 

15. 

-y/a'b'. 

20. 

^x^y^. 

WRITTEN   EXERCISES 


Simplify  the  following: 

3 


1.  Va?{a  -  by. 

2.  a/250. 

3.  ViiTaW^. 


4.    ^72x*yh^ 


5.  a/243  aV". 

6.  A/216a;Y(xTy7. 

7.  Va^  4-  5  a\ 


15.    a/64  6^06(^1". 


8.  Vl25(aj  +  yy. 

9.  a/18  a -9. 

10.  a/98  (a;  +  2/)2(a;  -  yf. 

11.  ^^^625  or*?/  (a;  —  yy. 

12.  A^aj^  -  2  ic3, 

13.  ^x'-^3a^b-\-3x^b^-\-xb\ 

14.  a/27  (a;  +  2/)(a- 6)3. 
16.    a/(x3  +  2  xV  +  ay)(ic-  +  6a;). 


ORAL  EXERCISES 

E-educe  the  following  to  entire  surds : 
7.  2a-\/x. 


1.  2Vx. 

2.  3a/2. 

3.  2^^. 

4.  2a^. 

5.  a-\/b. 

6.  aA/6. 


8.  a^A/aft. 

9.  2a^x. 
10.-  2^. 

11.  2a/2. 

12.  a6^. 


13.  ab^a. 

14.  xy^-vx. 

15.  a^^^A/o^; 


16.  a6x'2v2a6. 

17.  2  a2?7iS!/3  a?>i2. 


3/P 


1 8 .    ahnhi  V  2  a'^??!  7i^. 


266  KlUrilKK   OPKl^ATIONS   ON    KADIOALS 

WRITTEN    EXERCISES 

Koihu't'  tlu>  follDwins;-  ti>  t'litiro  surds: 

1.  'Jn  4.                          4.  L*.rS  .rv.  T.  (r\/h. 

2.  {n-\-b)  Va  +  b.          5.  'J.r-Vt^  .17/3.  8.  '2  inn\  {\  mn. 

3.  ;;\'2.                           6.   L*(rva«:  9.  (r7>  \  u^/,. 

10.  (jVoM^  21.   {a-\-h)Va-b. 

11.  5a•^/27a^.  22.   -ahc\'Jah7. 

12.  -^V(.T^  +  //^  +  2.ry/)xt/.  23    ,..yv^,Y^V 


13.   id  _. 


/^±T  24.   aSVa^+v'. 

14.  a/>^27^.  2^-   {.^^-.V)Vr+./. 

15.  4  xV'2  j^(/.  26.  abVab. 


16.  -2vv6iA 

^  — '  27.  (r(.r+  .  _,    , 

17.  (j-7''\  .iV-  '     >'rtV  +  .Vy 

18.  o.nr\  .r^/A 

19.  -2(fV'f76; 


28.    -M_J2+'l±r. 
20.   2.rv3\/:r^  29.    —  «?>('\/a?/c. 


190.  Fractional  Exponents.  Thus  tar  a  fra\'tiou  has  nt'ver 
lu'en  used  as  au  expout'iit,  aiul  I'viihMitly  it  I'ouUl  nt)t  be  so  used 
without  extendiiii;-  the  nu'aniug  of  the  word  cvponent. 

Tlui.s,  d"  means  a  •  a  ■  a,  buttc-  evidently  I'luumt  wwiwx  that  (t  is  to  be 
niultipHfd  by  iLself  Diie  haU"  of  a  time. 

We  shall  aijree  that  the  iiu'aiun^  attached  to  f'rmiionaJ  ex- 
j)onents  luust  be  sueh  as  to  make  thi'ni  obey  the  hiws  whieh 
govern  intvijnd  exponents. 

E.g.  Just  as  a-  •  a*  =  a^-^^  =  a^,  so  we  shall  agree  that  (ji  •  a-  must  e([u;ii 
oi-^*  =  a^  =  a. 


REDUCTION   OF    SURDS  267 

Since  we  agree  that  a-  •  a^  =  a,  we  see  that  a*  is  one  of  the 
tiL'o  equal  factors  of  a.     That  is,  a*  =  Vfl.     See  §  168. 

Similarly  a^  •  a^  •  a^  =  a^^^^'^  =  a.  Hence  a'  is  one  of  the  three 
equal  factors  of  a.     That  is,  a^  =^  a.     See  §  183. 

.  2  2  o  2.2.2  .  ., 

Again  a^  •  a^  •  a^  =  a^  ^  ^  =  a'  =  a: 

Hence,  a^  is  one  of  the  three  equal  factors  of  a-.     That   is 

2  3    — 

a    =  A  a-. 

Definition  of  Fractional  Exponent.  From  these  examples  we 
see  that 

Tlie  numerator  of  a  fractional  ejyponent  indicates  the  power  of 
the  expression  overvjhich  the  exponent  stands,  and  the  denominator 
indicates  ichot  root  of  this  power  is  to  be  taken. 

Thus,  ai  =  y/a^,  a^  =  \/a^,  x^  =  y/x,  x^  =  Vx,  etc. 

It  is  also  true  that  a^  =  V«*=  (Va)^  that  is,  either  the 
power  or  the  root  may  be  taken  first. 

E.g.     83  =  v/8^  =  V  64  =  4  ;  and  8  J  =  (  ^)2  =  2-  =  4. 

ORAL   EXERCISES 

Give  the  equivalents  of  the  following,  using  fractional  ex- 
ponents. 


1. 

V2. 
^3. 
^2. 

Vs. 

■\  2^. 

6. 
7. 
8. 
9. 
10. 

11. 

12. 
13. 
14. 
15. 

VaW. 
V  m^n*. 

16. 

17. 
18. 
19. 
20. 

^/abhj'. 

2. 

i^x'y'z'. 

3. 

■\'  xi^t/z'. 

4. 

y/nrn^if'. 

\^a=6\-«. 

5. 

A  xi/z^. 

S  m-n'^p'. 

Operations  with  Fractional  Exponents.  If  a  fractional  ex- 
ponent is  reduced  to  higher  or  lower  terms,  the  value  of  the 
whole  expression  is  not  changed. 

Thus,  4i  =  45,  since  4^  =  \  4  =  2,  and  4t  =  \/T^  =  v  64  =  2. 


268  FURTHER   OPERATIONS    ON   RADICALS 

191.  Reduction  of  Surds  to  a  Common  Index.  In  reducing  surds 
of  different  index  to  equivalQnt  surds  with  a  common  index, 
fractional  exponents  are  used  as  in  the  following  examples : 

1.  Reduce  Vo  and  VS  to  equivalent  surds  with  the  same 
index. 

jSolution.      Vl  =  b^  =  6^  =  'V^'=  V2E. 

2.  Reduce  Va  and  V6  to  surds  with  the  same  index. 
Solution,     va  =  a^  =  a^"  =  va*. 

3.  Reduce  -^x^  and  Vit*^  to  surds  with  the  same  index. 

3/ —  2.  4  Q. — 

Solution.  vcc-  —  x^  -  x''  =  vx*. 


y/x^  z=  x^  =  x^  =  Vx^. 
WRITTEN  EXERCISES 

Reduce  to  equivalent  surds  of  the  same  order : 

1.  Va,  Vb.  6.    V3,  Vs.  11.  a^,  Va,  Va. 

2.  Va,  V^.  7.    Vo-,  Va;.  12.  x^,  x^,  x^- 

3.  V2,  V2.  8.    ^/ab,  -^a^.  13.  2  Va,  3Va,  SVa. 

4.  V3,  a/2.  9.    Va^2,  V^.        14.  3  a^  2  a*. 

5.  V3,  V3.  10.    V5,  V25.  15.  2a;^  2a;^,  2x^^. 

ADDITION   AND   SUBTRACTION   OF   SURDS 

192.    Similar    surds    are    those  which    have    the    same    surd 
factors,  that  is,  the  same  index  and  same  radicand. 

E.g.  4v^  and  —  2y/a^  are  similar,  while  \/a  and  v^6  are  not  similar. 

Similar  surds  and  those  which  can  be  reduced  to  this  form 
may  be  combined  into  a  single  surd.     For  example : 

2.  \/32'  +  \/72  =^10^2  +\/36T2=4v'2+  6\/2  =  IOV2. 

3.  Vi-Vj=>/|^-\/iT3=  iVli-  .^\/3=-iV3. 


ADDITION   AND   SUBTRACTION   OF   SURDS 


269 


ORAL  EXERCISES 

Combine  each  of  the  following  into  a  single  surd: 

1.  3V5-2V5.  5.    2a/2  +  7a/2--J/2. 

2.  5a/2  +  3\/2.  6.    4:Va^  +  D^a^  -  S^a\ 

3.  2Vx  +  3Vx-Vx.  7.    12^^+8-</^-5-J/^. 

4.  a^b-cVb.  8.    aVb'^ -\- c^J?  -cl-\/'b\ 

WRITTEN  EXERCISES 

Simplify  each  of  the  following  as  far  as  possible  without 
approximating  roots. 

1.    V27  +  2V48-3V75.  2.    V20  +  Vl25  -  VM). 


3.   3V432-4V3  +  V147. 


4.    3V2450-25V2  +  4V13122. 


5.  3  yW.rh  +  2  Va;V  -  yz'J' 

6.  V4  0^?/  +  V25  a?//  —  x^/xy. 


xz 

2 


7.  v'«ic2  -  bx-  +  V4  ar^s^  —  4  fer^sl 

8.  4V|-fVA-3V27. 

9.  2V|  +  V60+V|.  10.    5V3-2V48  +  7V108. 

1.  Va^  -  ci'b  -  Va/;2  -  6^  -  V(a  +  6)(a2  -  52^. 

2.  Va  +  3 V2^ -  2^/3a  +  V4a  -  V8~a  +  Vl2a. 


3.    Va^  +  2  .^•22/  +  xy^  —  Va^  _  2  a;^?/  -|-  a^z/^  —  V4  xy\ 


4.  Vr-s+Vl6r- 16s+Vr^2^^s^  — V9(r^^). 

5.  ■\/(m  —  M)2a  +  V(w  +  nya  —  -Vam"  +  Vrt(l  —  m)^  —  Va. 

6.  V32  a^y"  +  Vl62  xY  -  V512  a^^t/*  +  V1250  xy. 

7.  ^16+^54-Sy250+^/l28. 

8.  \/54+ a/128 +Vi28-fV200. 

9.  v'32+ a/162  H--J/5l2-\/ 1250. 


270  FURTHER    OPERATIONS    ON   RADICALS 

MULTIPLICATION   OF   SURDS 

193.    Surds  with  the  same  index  are  multiplied  according  to 
§  187.  _  _ 

Va  •  "\/6  =  Vab. 

E.g.   </2.  ^l=-^/2^  =  ^8  =  2. 

Surds  of  different  index  but  with  the  same  radicand  are  best 
multiplied  by  use  of  fractional  exponents  (§  190). 

E.g.   V2  .  V2  =  23  .  2^  =  2^  +  2  =  2^=  V2^. 

Surds  of   different  index  and  having  different  radicands  must 
be  reduced  to  the  same  order  before  multiplying. 

E.g.   y/2  •  V3  =  23'  •  3^  =2^  •  3^  =  ^^2^  •  v'33=  \/'W^=  v^IT27  =  v/lu8. 

ORAL  EXERCISES 

Multiply  and  simplify  the  products  : 


1. 

V2. 

V3. 

7. 

V5  .  V7. 

13. 

V7 .  Vf 

2. 

</2. 

.^3. 

8. 

V6  .  Vl2. 

14. 

v|.vi. 

3. 

V7. 

V3. 

9. 

vi8.  V8: 

15. 

V|.V|. 

4. 

^'3. 

^9. 

10. 

Vx^  .  V.^"^ 

16. 

oi  .  2\ 

5. 

V2. 

.V8. 

11. 

Va'^b  '  Vab"^- 

17. 

3i  .  3i 

6. 

■^2. 

•  </l6. 

12. 

-VrsH  •  Vi-'s^f. 

18. 

4i  .  ii. 

WRITTEN   EXERCISES 

Multiply  and  simplify  the  products. 

1.  V6  .  V8.  7.  V6  .  A/i2. 

2.  V7  .  V^^.  8.  2V3.3^/2. 

3.  V^  •  VoP.  9.  2^16.  V4. 

4.  V20  .  V50.  10.  2  Vl2  .  Vl. 

5.  V5  •  Vl5.  11.  50^  .  20*. 

6.  Va-  a/6.  12.  2V7  .7V2. 


MULTIPLICATION   OF   SURDS  271 


13.  18^. si  ^^  (^2^YL^^^ 

14.    1003  .40i  .._.  1  ^ 


15.    3^.27^ 


18.  (ly .  (147)  • 


4, 


■     \  7/        \2  0^2*                        20.    (3  a6c)^  •  (6c)^  •  {a?hcf. 

194.    Binomial  Quadratic  Surds  are  multiplied  as  in  the  follow- 
ing example : 

3V2-4V5 
2V2  +  3\/5 

6-2  -SVTO 

+  9V10-  12  •  5 

1.  (V2-V3)2. 

2.  (1+V2)2. 

3.  (2-V3)2. 

4.  (1  +  V2)(1- 

12  +  VlO  - 
WRITTEN 

.V2). 

60  =  VlO  -  48 

EXERCISES 

5.  (V2-fV5)(V2-V5). 

6.  (Va+V6)(Va-V6). 

7.  V2(2V3+3V8-5V6). 

8.  -^abc{-^Ja-\--yJb-\-^c). 

9.  (3V5-2V3)(3V5H-2V3). 

10.  (1+V2)(1-V3  +  V5). 

11.  V3  +  V5xV3-V5=V(3+V5)(3-V5). 

12.  V4-2V3xV4  +  2V3. 

13.  V8  +  3V2xV8-3V2. 

14.  (7V5-2V3)(7V5+2V3). 

15.  (a  +  ^)(Va-V&)(Va+V6). 

16.  (3  a;  +  .r  V3)  (3  x  -  x  V3). 

17.  (V5+V2)(V5-V3> 


272 


FURTHER   OPERATIONS   ON   RADICALS 


195.    Powers  and  Roots  of  Monomial  Surds  are  best  fouud  by 
use  of  fractional  exponents,  as  in  the  following  examples : 

1.  (\/5)3  =(5^)8  =  6^  •  5^  .  5^  =  5  .  52  =  oVK 

2.  (23\/3)2  =  (23 .  v^)(23  .  \/3)  =  23  .  23  .  3^  .  3^ 

=  26.3^  =  26.3^  =64>/3. 

3.  (  v^3\/2)2  =  (3^  .  2^) (3^  •  2^)  =  3^  •  3^  •  23  •  2^  =  3^  •  2^  =  \/W^y/¥. 
In  general,  (a'"  •  b^y  =  a'""6'"'. 

In  the  above  formula  Ic  is  an  integer,  but  the  formula  is  also 
true  when  h  is  di  fraction. 

Thus  (V6i)^  =  J(64)^|^  =  64^  •  ^  =64^  =^64  =  2, 
which  we  know  to  be  the  value  of  (V64)3, 
since  (V64)3  ={/V64  =  "v^S  =  2. 

In  general,  v  Va  =  '"■v^fl- 

This  latter  formula   is   sometimes  useful  in  simplifying  a 
radical. 

E.g.   (1)    V^125=V\/l25=V5. 

(2)      v^=:V\/64=  >/8  =  V4T2  =  2\/2. 


ORAL  EXERCISES 

Perform  the  following  indicated  operations  : 


1. 

(0.^)^ 

10. 

(x^y. 

19.    ( 

>'¥)'. 

2. 

{x^y. 

11. 

(x^y 

20.    ( 

[x^yiy. 

3. 

(ah^y. 

12. 

(xfy. 

21.    ( 

,  2  1. - 

4. 

{ah^y- 

13. 

(xfy. 

22.    ( 

;a;V)«. 

5. 

(ai)3. 

14. 

(x^y 

23.    ( 

'cc^h^y. 

6. 

{a^y. 

15. 

(x^y 

24.    ( 

'cih^y. 

7. 

{ah'y. 

16. 

(x^y 

25. 

[ah^^y. 

8. 

{a'jy^y 

17. 

(x^y 

26. 

[ah^y. 

9. 

(4i)». 

18. 

(x^y. 

27. 

[ah^y\ 

MULTIPLICATION   OF   SURDS  273 


ORAL  EXERCISES 

3/7T  5/r>  6/- 


1.  Find  the  square  of  V5,   V2,    V3,   Va;. 

2.  Find  the  cube  of  V2,   a/5,   V2,   V^. 

3.  Find  the  4th  power  of  V3,   V3,    VS,   Va;. 

4.  Find  the  5th  power  of  V2,   V2a,    V3 a.',   V4 6. 

5.  Express  as  a  single  root  v  V2,    v  V5,    v  V5. 

6.  Express  as  a  square  root  V 8,    V36,   VlG. 

3  /  Q  . — -         3  /~4"7-—         3  /  g  / — ^ 

7.  Express  as  a  single  root  v  v  4,    v  v  3,    v  v  5. 

8.  Express  as  a  cube  root  V4,   V8,    v  16. 

9.  A  cube  root  of  a  fifth  root  is  what  root  ? 
10.  A  sixth  root  of  a  cube  root  is  what  root  ? 

WRITTEN  EXERCISES 

Perform  the  following  indicated  operations  : 

^a'^bf.  11.    (Va26V)l  21.    lx^yh^y\ 

SaWcf.  ^2.    [(a)^]i  22.    [a^^^j^^ 

^)'-  14.  [(2^)^ J.         24.  V^Tm 

^^/.  15.  [(a2)3]i.  25.    \/^. 

^<  16.  l(ax')^y.  26.    Va/64. 

^^)'-  17.  ^C^l/^.  27.    Va/2&6. 

a/^"^^)^-  18.  [(2/3)iji.  28.    VvlSg. 

^^^c^)'.  19.  [-y/ix  +  yfY-  29.    Va/T29. 

■^-  aWy,  20.  [a/(ci-6)2]^         30.    VV729. 

31.  </v^.  34.  [(27)^]^. 

32.  (</Sy'(^/iy  35.  (Vl2)3 .  (^I6)2. 

33.  V^.  36.  l\^xy(a-b)\'^Y. 


1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 

9. 

10. 


274  FURTHER   OPERATIONS   ON   RADICALS 

DIVISION  OF   SURDS* 

196.  When  a  divisor  is  a  surd,  it  is  convenient  to  indicate  the 
division  in  the  form  of  a  fraction  and  then  to  reduce  the  de- 
nominator to  the  rational  form  as  in  the  following  example: 

V5      ^       V5(V5+V2)       ^5+Vio^5  +  Vio 
V5-V2      (V5-\/2)(V5-hV2)        5-2  3 

Explanation.  We  see  that  if  we  multiply  the  denominator  by  \/5+  \/2 
we  shall  have  the  product  of  the  sum  and  difference  of  \/b  and  V2,  which 
is  equal  to  the  difference  of  their  squares. 

That  is,  (\/5)^— (\/2)-=  5  —  2=3,  which  is  rational. 

If  now  we  multiply  the  numerator  also  by  VE  +  \/2,  the  value  of  the 
fraction  will  not  be  changed.     The  numerator  then  becomes 
y/l{y/l  +  V2)  =  (>/5)2  +  .V5  .  V2  =  5  +  VlO. 

The  success  of  this  proceeding  depends  upon  choosing 
Vo  +  V2  with  which  to  multiply  the  denominator  so  as  to 
obtain  a  rational  j^roduct. 

197.  Rationalizing  the  Denominator.  This  is  called  rational- 
izing the  denominator,  and  the  factor  by  which  we  multiply  is 
called  the  rationalizing  factor.     See  §  178. 

If  we  wish  to  compute  the  approximate  value  of  V5-f-(\/5—  V2) 
above,  it  obviously  requires  less  numerical  work  to  use  the  form 
(5+ VlO) -^3  with  rational  denominator,  instead  of  VS -=- ( VS  —  \/2) , 
which  involves  the  extraction  of  two  square  roots  and  a  long  division,  while 
the  former  requires  the  extraction  of  only  one  root  and  a  short  division. 

To  rationalize  the  denominator  of  a  fraction,  it  is  necessary  first 
to  find  an  expression  which,  multiplied  by  the  denominator  of 
the  fraction,  gives  a  rational  product,  and  then  to  multiply  both 
terms  of  the  fraction  by  this  expression. 

In  the  case  of  a  monomial  or  binomial  quadratic  surd,  the 
rationalizing  factor  may  be    found  at  sight. 

Thus,  if  the  denominator  is  of  the  form  V.r  or  a^/x,  then  Vx  is  the 
rationalizing  fiictor,  since  Vx  ■  y/x  =  x. 

If  the  denominator  is  of  the  form  Vx  +  \///,  then  Vx  —  Vy  is  the 
rationalizing  factor,  since  ( Vx  +  Vy){Vx  —  Vy)  =x  —  y. 

*  Articles  196  and  197  may  be  omitted  without  destroying  the  continuity. 


DIVISION   OF   SURDS  275 

ORAL  EXERCISES 

Give  a  rationalizing  factor  of  each  of  the  following: 


(1)  V3a;,  (2)  V2a!36,         (3)   V8  6V,         (4)  Va  +  V6, 

(5) Va  - V6,  (6)  V3a  +  V2h\  (7)  a/7  -  V27; 

ORAL  EXERCISES 

Rationalize  the  denominators  of  each  of  the  following : 
1.    ^.  5.    _^ •         9.  1 


1 


Vx  4- 1  V7  —  V3  V^  —  V.V2 


V3  +  V5 

2 

V7-V3 

2 

Va  +  1 

1 

10. 


1 7  2  11    V2  +  V3 

V2-1  ■  '   Va  +  1  '         V2 

1.1  12    V8-V3 


V5-V3                Va  +  V&  V3 

WRITTEN  EXERCISES 

Rationalize  the  denominators  of  each  of  the  following : 

^^    V^  +  Vy.             7.   ^+^    _.  13^      2V5  +  1  . 

V^                       aVx  —  bVy  '   3V5— V3 

2    3V6H-9V2         8.   ?-+^.  14    6V3  +  4V2 

2V2                     3-V2  '  6V3-4V2 


4. 


7 

a-\-b 

•  • 

aVa?  —  6V2/ 

8 

2  +  V3 

3-V2 

9 

5  +  V6 

4-V3 

10. 

2-V5 
3  +  V3 

11. 

a  +  V5 
a  — V6 

^9.. 

a  — a/5 

3  V6  9.   5  +  V6  ^^     A/a;-3-3 

A/a  +  V6  4-V3  '    V^^^  +  3 


-s/a  —  Vb  1ft  ^- v5  a;_Va  +  6 

Va  +  v6  3  +  V3  "  X  +  Va  +  6 

g     Va4-V5  11.  a±V5.  17^  1  +  V2  +  V3 

Va-V6*  '  «-V&  *  l-V2-hV3 

6.          J  12.  ^-^  18.  5  +  V2-V3 

aVa;  +  &V.v  a4-V6  5+V2+a/3 


276  FURTHER   OPERATIONS   ON   RADICALS 

EQUATIONS   mVOLVmG    RADICALS 

198,    Illustrative  Example.     Solve  the  equation : 


Vx-  5  +  Vx  +  1  -  3  (1) 

By  S\Vx+l                          \/x-5  =  3-Vx  +  l  (2) 

Squaring  both  sides                     x  —  6  =  9  —  6\/x  +  1  +  x  4-  1.  (3) 
Transposing,                            6Vx  +  1  =  15 

or  2Vr+T  =  5  (4) 

Squaring  both  sides  .     4(x  +  1)  =  25  (5) 

Hence  x  =  51  (6) 

Check.     Substitute  x  =  5|^in  (1) 

If  two  radicals  are  involved,  or  one  radical  and  one  or  more 
rationar  terms,  it  is  best  to  get  a  radical  alone  on  one  side  of 
the  equation  before  squaring  as  in  Equations  (2)  and  (4). 
Note  that  squaring  both  members  of  an  equation  is  equivalent 
to  multiplying  both  sides  by  the  same  number. 

WRITTEN  EXERCISES 

Solve  the  following  equations  and  check  each  result. 


1.    Va;-5  =  3.  7.    V4.i-2-7  +  2ic  =  7. 


2.    X  — 2=^x^  —  4:.  8.    Va;  +  1  —  Va;  —  7  =  2. 


3.    Va?  +  6  =  3Vx  -  2.  9.    V//  +  4  +  V?/  —  1  =  5. 


4.    ^x^  —  2x-\-S  =  x—4:.  10.    V.x- +  2  =  Vx  +  16. 


5.    Vaj2+7a;— 4=Va;2+8a;— 5.    11.    5  —  Vx  =  V'a; -f  5. 


6.    Vx2  +  5  +  x  =  5.  12.    Vx  +  7=:1 -f-Vx  +  2. 


V^  +  1     Vx  +  13  Vx+1     Vx-1     ^-1 


14     Vfl;-2^  Vx+l  jg     2-Vx^3  + Vx. 

Vx  — 4      Vx"-^  1  H-  Vx      3  —  Vx 

Suggestions.     In  Example  13  clear  of  fractions  first ;  in  14  square  both 
members  and  then  clear  of  fractions  ;  in  16  rationalize  the  denominators. 


REVIEW   QUESTIONS  277 

REVIEW   QUESTIONS 

1.  Give  examples  of  rational  expressions,  of  surds,  of  quad- 
ratic surds. 

2.  What  three  reductions  of  surds  are  considered  in  this 
chapter  ? 

3.  How  is  Principle  XVIII  used  to  reduce  a  mixed  surd 
to  an  entire  surd  ? 

4.  State  in  words  the  extension  of  Principle  XVIII  which 

in  symbols  is : 

Va  •  b  =  \^a  •  -\/b. 

5.  How  is  a  fractional  exponent  defined  ?     Show  by  an 

example  such  as  16*  x  16^  that  the  law  of  exponents, 

holds  for  fractional  exponents. 

6.  How  are  fractional  exponents  used  in  reducing  surds 
with  different  indices  to  equivalent  surds  with  the  same  index  ? 

7.  What  are  similar  surds  ?  When  can  two  or  more  surds 
be  combined  into  a  single  surd  by  addition  or  subtraction  ? 

8.  By  what  principle  are  surds  of  the  same  index  multi- 
plied? Surds  of  different  index  but  with  the  same  radicand? 
Surds  of  different  iudex  and  having  different  radicands  ? 

9.  State  in  words  the  principle : 

(a'"  •  a"Y  =  a'""  •  6^". 

10.*    How  do  we  divide  by  a  binomial  quadratic  surd  ? 

11.*  Give  examples  to  show  how  a  fraction  with  a  binomial 
surd  denominator  may  be  changed  into  an  equivalent  fraction 
having  a  rational  denominator. 

12.  In  solving  an  equation  containing  radicals,  how  are  they 
removed  (1)  when  only  one  radical  expression  is  invoh  ed  ; 
(2  /  when  two  such  expressions  are  involved  ? 


CHAPTER   XVIII 
QUADRATIC  EQUATIONS 

199.  Equations  of  the  form  x"^  -{-  ax  +  b  =  0  have  already 
been  solved  in  cases  where  the  left  members  could  be  factored 
by  inspection.  See  §  116.  However,  in  a  case  like  x^  +  5x 
4-3  =  0,  the  factors  of  the  left  member  cannot  be  found  by 
any  method  thus  far  studied.  It  is  therefore  necessary  to  con- 
sider other  methods  for  solving  equations  of  this  type. 

200.  Completing  the  Square.  As  a  preliminary  step  we  con- 
sider again  the  properties  of  a  trinomial  square.     See  §  97. 

From  (x  -\-  ay  =  x"^  +  2  ax  +  a^  we  see  that  the  third  term  is 
the  square  of  half  the  coefficient  of  x  in  the  second  term. 

Hence,  if  we  had  given  only  the  two  terms  x^  +  2  ax,  we 
would  have  to  add  a^  in  order  to  cornplete  the  square. 

E.(j.  To  find  the  term  to  be  added  to  x-  +  6x  in  order  to  complete  the 
square,  we  take  the  square  of  half  the  coefficient  of  ic,  that  is  3-  =  i). 
Thus,  a;2  +  6  X  +  9  is  a  complete  square. 

In  general,  to  complete  the  square  in  x-  -f-  px,  we  add 
f-p  )  =— .     Thus  x'^-\-px-^^  is  a  complete  square. 

ORAL   EXERCISES 

Complete  the  trinomial  square  in  each  of  the  following : 

1.  x^-\-2x.                    4.    x'-\-^x.  7.    (3.T)2  +  2(3.r). 

2.  ^-\-\x.                    5.    ic2  +  3a;.  8.    (2.t)-  + 4(2a'). 

3.  ^-\-^x.                     6.    x'^-bx.  9.    l()a-2  +  2(4x). 

10.  How  do  you  complete  the  square  in  or  —  2ab?  Is  the 
rule  different  in  this  case  ? 

11.  Complete  the  square  in  each  of  the  above  exercises,  after 
replacing  the  sign  -|-  by  — . 

278 


SOLUTION   BY   COMPLETING   THE   SQUARE  279 

201.     Solution  of  a  Quadratic  by  Completing  the  Square. 

Example  1.     Solve  the  equation:  a;^  +  6 a?  +  o  =  0.  (1) 

Transposing  in  (1),  x^  +  6  x  =—  6.  (2) 

To  complete  the  square,  we  add  3^  =  9  to  both  members. 

Thus  we  have  :x;2  +  6  x  +  3-  =  S'-^  -  5  =  4.  (3) 

Taking  square  roots  of  both  sides,   x  +  S  =±  ■\/4  =  ±  2. 

Hence  x  =—  S  +2=—  1, 

and  a;=— 3  —  2=—  5. 

Example  2.     Solve  the  equation  : 

a;2- 12x +42  =  56.  (1) 

Transposing,  x^  —  l'2x  =  14.  (2) 

Completing  the  square  by  adding  (-/)2  =  B"-^  =  36  to  both  sides, 

x-2  _  12  X  +  36  =  14  +  36  =  50.  (3) 

Taking  square  roots,  x  —  6  =  ±  VbO  =  ±  5>/2.                (4) 

Transposing,  x  =  6  ±7.071.                            (5) 

Hence  x  =  6  +  7.071  =  13.071, 

and  also  x  =  6  —  7.071  =  —  1.071. 

The  steps  involved  in  the  above  solutions  are : 

(1)  Write  the  equation  in  the  form  x^  -+  px  =  q. 

(2)  Complete  the  square  by  adding  {\pY  to  each  member. 

(3)  Take  the  square  root  of  both  members  of  this  equation. 

(4)  Solve  each  of  the  first  degree  equations  thus  obtained. 

WRITTEN  EXERCISES 

In  solving  the  following  quadratic  equations  the  result  may 
in  each  case  be  reduced  so  that  the  number  remaining  under 
the  radical  sign  shall  be  2,  3,  or  o.  (§  180.)  Use  V2  =  1.414, 
V  3  =  1.732,  V5  =  2.236. 


1. 

a;2  _4:c  =  8. 

6. 

a:2 

-12x  =  12. 

11. 

8  =  .1-2  -h  4  X. 

2. 

if2  =  3-6ic. 

7. 

x" 

-So.' =  -14. 

12. 

23-6  X  =  x\ 

3. 

4  X  =16-  x\ 

8. 

x"- 

=  2^- -hi. 

13. 

7  4-  2x  =  x'. 

4. 

x^  +  6  X  =  9. 

9. 

.i'2 

-  4  a;  =  16. 

14. 

25  -  x2  =  5  X. 

5. 

x''''  +  iSx  =  11. 

10. 

.1-2 

==23+-4.c. 

15. 

x^  -{-lx  =  2. 

280  QUADRATIC   EQUATIONS 

202.  The  Hindu  Method  of  Completing  the  Square.  In  case  the 
coefficient  of  x^  is  not  unity,  as  in  3  ic^  +  8  a;  =  4,  both  members 
may  be  divided  by  this  coefficient,  and  the  solution  is  then 
like  that  of  Examples  1  and  2  on  page  279. 

However,  the  following  method  is  sometimes  desirable : 

3  0:2 +  8  a;  =  4.  (1) 

Multiplying  each  member  of  equation  (1)  by  4  •  3  =  12,  we  get 

36  a;2  +  06  a;  =  48.  (2) 

This  can  now  be  written  in  the  form  x"^  -f  px  =  g,  namely, 
(6x)2  +  16(6 x)=  48,  in  wliich  p  =.  \Q  and  6x  is  the  unknown.  Hence, 
we  add  (V)"^  =  8^  =  64  to  complete  the  square,  and  get 

(6 xy  +  16(6  X)  +  64  iiz  48  +  64  =  112.  .      (3) 

Taking  square  roots,  6  x  +  8  =  ±  Vll2  =  ±  4V7.  (4) 

Hence  6x=-8±4\/7, 

and  a:=— fifV?.  (5) 

In  this  solution  both  sides  were  multipUed  by  4  times  the  original  coeffi- 
cient of  a;2,  and  then  the  number  added  to  complete  the  square  was  found 
to  be  the  square  of  the  original  coefficient  of  x. 

The  advantage  of  this  form  of  solution  is  that  fractions  are 
avoided  until  the  last  step,  and  the  number  added  to  complete 
the  square  is  equal  to  the  square  of  the  coefficient  of  x  in  the 
original  equation. 

This  is  called  the  Hindu  method  of  completing  the  square 
because  it  was  first  used  by  the  Hindus. 

Note.  —  Fractions  would  also  be  avoided  in  the  above  solution  if  equa- 
tion ( 1 )  were  multiplied  by  3  instead  of  4  •  3.  This  is  the  case  only  when 
the  coefficient  of  x  is  an  even  number. 

Hence,  if  the  given  equation  is  in  the  form  ax"^  +  bx  -f-  c  =  0, 
we  may  complete  the  square  without  dividing  through  by  a,  by 
the  following 

Rule.    1.    Write  the  eqitntion  in  the  fmmi  ax"^  -\-  bx  =—  c 

2.  Multiply  both  sides  by  4  a. 

3.  Jfow  lurite  the  equation  in  the  forjyv 

(2  axf  +  2  6  (2  fljr)  =  -  4  ac. 

4.  Complete  tJie  square  by  adding  b"^  to  both  sides. 

5.  Solve  as  though  2  ax  were  the  unknown. 


CHECKING   QUADRATIC   SOLUTIONS  281 

EXERCISES 

In  the  solution  of  the  following  equations  the  roots  which 
contain  surds  may  be  left  in  simplified  radical  form. 

1.  2a;2  +  3a;  =  2.  11.  2x^-{-4:X  =  23. 

2.  3  a;2  +  5  a;  =  2.  12.  3  a;^  _  7  =  4  x. 

3.  3  a;  =  9  -  2  a;2.  13.  2  x""  -  5  =  3  a. 

4.  6x-j-l=-3x\  14.  4  a;2  =  6  a;  -  1. 

5.  2  a;2  =  5  a;  +  3.  15.  2  x  =  1  -  5  x\ 

6.  4.x  =  2  x""-!.  16.  3  a?  -  20  =  -  2  x\ 

7.  2a;2-3a;  =  14.  17.  2  a^ -f- 3  a;^  =  9. 

8.  3  a:2  =  9  +  2  a;.  18.  4  3^2  -  1  =  3  a;. 

9.  4  a;2  =  2  a;  +  1.  19.  4  a;  ==  7  -  2  x\ 
10.  6x~l=8x\  20.  2a;  +  l  =  5a;2. 

CHECKING   RESULTS  IN   QUADRATIC   EQUATIONS 

203.  Illustrative  Examples.  1.  Solving  a;^  —  7  a;  -|-  12  =  0,  we 
get  a^  =  4  and  x  =  3. 

What  is  the  sum  of  these  roots  ?  How  does  this  compare 
wdth  the  coefficient  of  x?  What  is  the  product  of  these  roots? 
How  does  this  compare  with  the  known  term  of  the  equation? 

2.    Solving  a;2  —  6  a; +  4  =  0,  we  get  a.'=3+ Vo  andx=3— Vo. 

The  sum  of  these  roots,  (3  +  Vs)  +  (3  —  \/5)  =  6,  is  equal  to  the  coeflB- 

cient  of  x  with  the  sign  changed;  and  the  product,   (3  H- 'n/5)(3  —  V5) 
=  9  —  5  =  4,  is  equal  to  the  known  term  of  the  equation. 

EXERCISES 

Solve  the  following  equations.  In  each  case  compare  the 
product  of  the  roots  with  the  known  term  and  the  sum  of  the 
roots  with  the  coefficient  of  x. 

1.  a;2-5a;  +  3  =  0.  4.    a-2-4.r-8  =  0. 

2.  a;2  +  3a;4-2  =  0.  5.    .^H- 6a;- 3  =  0. 

3.  a;2  +  9a;  +  8  =  0.  6.   a;2-8a;  =  6. 


282  QUADRATIC   EQUATIONS 

204.  Relation  of  Roots  and  Coefficients.  These  exercises  are 
illustrations  of  a  general  rule  for  all  quadratics  written  in  the 
form  x^  -{-  px  -{-  q  =  0,  in  which  the  coefficient  of  the  squared 
term  is  +  1 ;  namely  : 

TJie  sum  of  the  roots  is  equal  to  the  coefficient  of  x  with  its  sign 
changed,  i.e.  — /?;  and  the  ijroduct  of  the  roots  is  equal  to  the 
known  term,  q. 

This  may  be  used  to  check  the  results  obtained  in  solving  a 
quadratic.  Note  that  before  applying  the  test  the  equation  must 
be  in  the  form  specified.     See  example  on  next  page. 

205.  Solution  of  the  Quadratic  by  Formula. 

Solve  the  equation 

ax^  ■\-  bx  +  c  =  0.  (1) 

Transposing,  aoi?  +  hx  =  —  c. 

Multiplying  by  4  a,  4  a%^  +  4  ahx  =  —  4ac.  (2) 

Equation  (2)  may  be  written  in  the  form 

(2  ax)-^  +  2  5(2  a:c)  =  -  4  ac. 
Completing  the  square  as  if  2  ax  were  the  unknown, 

(2ax)2  +  2  6(2ax)+ &•-=  &--4ac.  (3) 

Taking  square  roots,  2ax  +  b  =±  Vb'^  —  4:ac.  (4) 

Transposing,  2ax  =—  b  ±  Vb^  —  4 ac. 

■r^-  .J-      1     «  —b±Vb'  —  ^ac       /CN 

Dividmg  by  2  a,  x  = ^— .      (5) 

2a 

Calling  the  two  values  of  x  in  the  result  Xi  and  Xo  we  have, 


6+  V62-4ac  _6-V62-4 


Xi  — ;  X2  — 


ac 


2a  '     '  2a 

Any  quadratic  equation  may  be  reduced  to  the  form  of  (1) 
by  simplifying  and  collecting  the  coefficients  of  x"^  and  x. 
Hence  any  quadratic  equation  may  be  solved  by  substituting 
in  the  formulas  just  obtained. 


SOLUTION   OF  THE    QUADRATIC   BY  FORMULA       283 

Example.     Solve  2x^  — 4x-{-l  =  0. 

Substituting  a  =  2,  b  =  —  i,  c  =:  1  in  the  formula, 


we  get  (-4)^V(-4)^-4.2.1, 

^  2-2 

From  which  x-\  =  — and  Xo  = 

2  2 

Check.     Writing  the  equation  in  the  form  x-+px-\-  g  =  0,  we  have 
a;2  —  2  X  +  ^  =  0,  in  which 7)  =  —  2  and  q  =  \. 

Then  iKi  +  ^2  =  — ^^ +  — ^ =  2^         "-^' 

„,                                            2+\/2      2-V24-2, 
and  Xi  •  2:2  =  — ^^^ •  = =1=0.- 


EXERCISES 

Solve  the  following  equations  by  the  formula  and  check  all 
results  by  means  of  §  204. 

1.  4  3.-2  +  1  =  8a;.  16.  8  + 4a;  =  3^2. 

2.  2x^  -'^x=  20.  17.  10  +  4  a;  =  5  x\ 

3.  2a;2-3=-5a;.  18.  2  +  oa;=3a;2. 

4.  3a;2  +  4a;=8.  19.  3  a;  +  14  =  2  a;2, 

5.  10  -  4  a;  =  5  or*.  20.  3  a;^  -  2  a;  =  5. 

6.  l  +  4a;2  =  -6a;.  21.  2a;2  +  4a-  =  l. 

7.  5  -  3  a;  =  2  a;2.  22.  4  x"  +  3  a;  =  1. 

8.  7  +  4a;=2a;2.  23.  2  a:^- 4  a;  =23. 

9.  6  a;2  +  12  a;  =  2.  24.  2  x"  -  3  x  =  -  1. 

10.  6a;2-12a;  =  - 2.  25.  3  .r  +  9  =  2  x^. 

11.  6 0.^  +  12 a;  =  -2.  26.  ox"  +  lx  =  7. 

12.  6a;2-12.T  =  2.  27.  2  .7;  -  1  =  -  4  .r^. 

13.  3  .x'2  4-  2  a;  =  5.  28.  5  x"  +  16  x  =  -  2. 

14.  2  +  3a;  =  2.i'2.  29.  6  .1-2  +  11 . 1- =  10. 

15.  8a;  +  l=-4a;2.  30.  5  o.-^  -  11  a:  -  12  =  0. 


284  QUADRATIC   EQUATIONS 

IMAGINARY  NUMBERS* 

206.  There  are  quadratic  equations  which  have  no  roots 
expressible  in  terms  of  the  numbers  of  arithmetic  or  algebra 
thus  far  studied. 

Example.     Solve  a;^  —  4  a?  =  —  8. 

Completing  the  square,      cc^  —  4a;  +  4  =  — 8+4=  —  4. 
Taking  the  root  and  transposing,       x=  2  ±  V  —  4. 


V— 4  is  thus  far  unknown  to  us  as  a  number  symbol,  but  we 
will  now  enlarge  our  number  system  by  including  in  it  numbers 
of  the  type  V— 4. 

207.  Definition  of  Imaginaries.  An  even  root  of  a  nega- 
tive number  is  called  an  imaginary  number.  All  other  numbers 
are  called  real  numbers. 

Using  Principle  XVIll  we  may  reduce  an  imaginary  number 
to  the  standard  form  a  V—  1,  in  which  a  is  a  real  number. 


E.g.  V-4=\/4  x(- 1)  =  V4.  V- 1  =2V- 1. 


V-5  =  \/5  x(- 1)  =  V5.  V- 1. 

208.    The    Fundamental    Operations    on    Imaginary    Numbers. 

In  operating  upon  imaginary  numbers,  they  should  first  be 
reduced  to  the  standard  form  aV— 1,  and  then  treated  as  in 
the  following  examples  : 

(A)  Addition  and  Subtraction. 

(1)  v^^+V^^T6  =  V4V^n[+\/T6\/^^ 

=  2  V^l  +  4\/^^  =  ()\/^^. 

(2)  \/iry  _  v^5  =  vr)  \/^n^  -  vs  v^T 

=  3  v^n[  -  vsv^n:  =(3  -  \/5)  v^n:. 

Thus,  in  general,  we  have 
*  Articles  206-209  may  be  omitted  without  destroying  the  continuity. 


IMAGINARY    NUMBERS 


285 


(B)  Multiplication  of  imaginaries  is  based  upon  the  principle 
that  the  square  of  the  square  root  of  a  number  is  the  number 
itself 

Thus,  (V'^n.y2=-i. 

Applying  this  principle  again,  we  have 

(V=T)3=(V^T)2.  V^^=  -  1  •  V^T=  -v^i;. 
And  still  again, 

(v/^T)*=(V:^l)2.(V^i)2=(_i)(_i)  =  +i. 

The  results 

(V^iy  =  -  1 ;  {V^^y  =  -  V^=^  ;  ( V^l)*  =  1, 
should  be  remembered.     They  are  used  in  performing  opera- 
tions like  the  following : 

(1)  V^^-  V^^  =\/2V^l.  \/8\/31  =V2\/8(\/^l)2 

VT6(-  l)  =  -Vl6=-4. 

(2)  V^l.  V^^- >/^l6=V4.  VO-  \/l0.(V^^)3 

^2.3.  4(-\/^l)  =  _24>/^n. 

(3)  V^4  .  V^)  ■  V^l6  .  V^2^  =  2  .  3  •  4  .  5(  V^l)^ 

=  120(+  1)=  120. 

In  general,  two  imaginary  factors  give  a  negative  real  product, 
three  imaginary  factors  give  a  negative  imaginary  product,  and 
four  imaginary  factors  give  a  positive  real  product. 

(C)  Division  by  an  imaginary  number  is  best  indicated  in 
the  form  of  a  fraction,  as  in  the  following  examples : 


(1)  2-V-3=-4— =     

a/_3      V-3  . 


2V-3 


2V~- 


(2)      V-9--\/-16  = 


-3      (\/-3)2 
V9  •  V^^l       3     - 


2x^  =  _2vi:3. 

-3  3 


Vie.  V- 1    4    y/zTi 


1^3    J  ^3 


EXERCISES 


4+V-9-V-16. 


Aadition  and  Subtraction 

1.  V 

2. 
3. 


V-25-V-9-V-4. 


4. 
5. 
6. 


V-4  4-V^^-V-16. 


V— 5  4-4V— 5  -h  TV—o. 


V-3-3V-3-4V-3. 


286  QUADRATIC   EQUATIONS 

EXERCISES 

Multiplication  and  Division. 


1.    V-l-V-1.  7.    V-16.V-9. 


4.  (V^^)^  10.  V^^^V^^. 

5.  V"=^-V^^.  11.    V^^-^(V^^)^ 


6.    V-2.V-3.  12.    (V-5)2--(V-2)^ 


13.    Multiply  2\/-  2  -  3V-  3  by  3V-  2  -  2V-  3. 

Solution,  2  V^^  -  3  V^^ 

3^172 -2V^^ 

6(V^^)2  -  9V2  v/3(V^n)2 

-  4V2 V3(\/^^)2  +  6(\/^r3)2 
6(- 2)- 13V6(- 1)  +  6(- 3) 
Simplifying,  -  20  +  13\/6. 

14.  (2-V^=^)(2  4-V^^). 

15.  (V^^+V'^^)(V^=^-V^^). 

16.  (2  -  3  V^^)  (2  -  5  V^. 

17.  (3-V^^)(3-4V=^). 


18.  (2+V-2)2.  20.   (V-3+V-2)2. 

19.  (;3-V'^)2.  21.  (V^^-V^^)2. 

209.    We  may  now  show  that  the  values  of  x  found  in  the 
example  of  §  206  do  actually  satisfy  the  equation. 

(1)  Substituting  x  =  2  +  V—  4  in  the  equation  a:^  —  4  x  =  —  8,  we  have 
(2  +  \/^^)"^  -  4(2  +  V—  4),  which  should  reduce  to  —  8. 

Squaring  2  +  V^^,  we  have  22  +  2  •  2  V^^  +  ( V^^)"'^. 
Simplifying  this,  we  get  4  +  4\/—  4  —  4  =  4.y/ —  4. 
Finally,  subtracting  4(2  +  V—  4),  we  have 4 V—  4,—  8  —  4V— 4  =  —8. 
Hence  the  left  side  of  tlie  equation  reduces  to  —  8  when  2  4-V—  4  is 
substituted  for  x,  and  the  given  equation  is  satistied. 

(2)  In  the  same  manner  show  that  2  —  \^—  4  satisfies  the  equation. 


QUADRATICS   IN   FRACTIONAL  FORM  287 

Quadratics  with  Imaginary  Roots. 

Solve  the  following  equations.     Simplify  each  result. 

1.  7  -  3x  =  -5x^.  7.  a;2  +  8  +  3 .^•  =  -  x. 

2.  11  a: -33  =  3.^2.  8.  ll.T2-49.T-f57  =  0. 

3.  14a?  +  8-a-2  =  52  +  3a;2.  9.  3a;2  +  18  -  12;^- =  o. 

4.  12 -ir).i-  =  ;^()-hGa;2.  10.  .37  -  4  ;r2  -  12.r  =  79. 

5.  5  X  +  if2  -f  8  =  0.  11.  10  .^2  +  46  +  7  a;  =  44. 

6.  5x'--10x  =  -6.  12.  45  +  oa;2-2a:  =  0. 

FRACTIONAL  EQUATIONS   LEADING   TO   QUADRATICS 

Solve  the  following  equations  and  check  each  solution  by 
substituting  in  the  original  equation,  except  when  the  answer 
is  given  : 

^     3a;- 1      4a; +  3         x^     ^         27      ^ 
x-\-l         x—1       x^  —  1  x^  —  1 

2.    3^+j      2^+1^        a;-l ^,,.,;=.|,orl. 

a;_9        a;  +  2       ic2-7a;-18  "' 


3. 


x-4:        3a;-15  3»2-114 


2a;-10       2a;-6  4a;2-32a;  +  60 

^     6fe+41_3(2a;-l)^7.  ^...  a;  =  1,  or  - 6i. 

x+5  x+1         2 

^     'dx  —  4:  ,  5x  —  7           9a;2  —  38  ^  ioi         o 

5. = Ans.  X  =  184,  or  2. 

x-4.       2x-2      2a;2-10a;  +  8  ^ 

x  —  2      3  —  a;      a;— 4      x—  6 

7     ^  +  2      3a;-15^3a;-21 

X  —  5        X  —  3  X  —  3 

_     2a;-3  ,  3x4-1      4a;-fl7        .        ,      -19±V345 

o. 1 ^  •       j!i.llS.  X  — 

—  4  a;        a;  —  2  a;— 2  4 

^     3a;-2      2a;2  +  15a;4-28  ,  2x-l 


10. 


2a;-t-3        2a.'2  +  5.c  +  3         x  +  1 

2a;-3       a;-8  ^    a;  +  2 
2x  +  2      5a;  +  2~  2a;  +  2  * 


288  QUADRATIC   EQUATIONS 

MISCELLANEOUS   QUADRATICS 

Solve  as  many  as  possible  of  the  following  equations  by  fac- 
toring. Otherwise  use  the  formula  of  §  205.  In  the  case  of 
surd  solutions,  find  the  results  correct  to  two  places  of  deci- 
mals. 

1.  a;2  +  llx  =  210.  21.   2a;2+3a;-3  =  12  » -f- 2. 

2.  50.-2- 3a;  =  4.  22.    3 a.-^  -  7 a;  =  10. 

3.  7a;  +  3ar^-18  =  0.  23.    17  a; -}- 31  +  2ar^  =  0 

4.  2  =  5x  +  7x\  24.   lS-4.1x  =  3-\-:^. 

5.  6a;-llx2^-7.  25.   10a; +  25  =  5  -  2a;  -  a^l 

6.  _5l4-42a;-3a;2  =  0.  26.    3a;  -  59  +  a;^  =  0. 

7.  3a;2  +  3a;  =  2a;-|-4.  27.    5a;2  +  7  a;  -  6  =  0. 
8.*    13-8a;  +  3a;2  =  0.  28.    a;2  +  12  =  7a;. 

9.  2a;2+lla;=32a;-a;2-27.  29.    8a;-5a;2  =  2. 

10.  176  +  3a;-a;2  =  2a;.  30.    5a;  +  3a;2  -  22  =  0. 

11.  a;2  +  6a;-54  =  0.  31.    50 -f  20  a;  +  ir^  =  5  a,-. 

12.  5x-2+9a;  +  12  =  4a;2+a;.  32.=*   a;2 -f  a;  +  4  =  0. 

13.  2a;2-4a;-25  =  0.  33.   20a;-|-2a;2+42  =  33a;+a;2^ 

14.  7a;2  +  lla;-6.  34.    17  a;  -  3^;^  =  -  6. 

15.  2a;2- lla;  +  5  =  0.  35.    8a;  +  5a.-2  =  -2. 

16.  2a;2-lla;  =  6.  36.    10  +  15 a;  +  a;^  =  26 a;. 

17.  25a;-95  =  a;2.  37.    3a;2-2x-7  =  0. 

18.  lla;2-42a;  =  2.  38.  5ar' -  9a;  -  18  =  0. 

19.  a;2-8a;-4  =  a;-22.  39.    7a;- 7  .x'^  +  24  =  0. 
^0.*    8a;2  +  5a;  =  -8.  40.*    31  +  2  a;  +  a.-2  =  0. 

41.  7 a;2  +  7x  -  5a;2  +  20  =  a;2- 2 a; +  2. 

42.  o-T^-f  3a;-7=(a;-l)(a;  +  2). 

43.  (a;-3)2-(2a;-l)(2a;  +  l)+7  =  0. 
44.*   3a;+(3a;-2)2  =  4a;2-l. 

45.  9a.'2-(2a;-l)2  =  (a;-f-3)2. 

46.  7a;2  =  5a;-(a;- 2)2  +  7. 


EQUATIONS   IN   THE    QUADRATIC   FORM  289 

EQUATIONS   IN   THE  QUADRATIC  FORM 

210.  Special  Cases.*  Sometimes  equations  of  a  higher  degree 
than  the  second  may  be  solved  by  means  of  quadratics,  as  in 
the  following  examples : 

Example  1.     Solve  a;^  -  or^  -  12  =  0. 

Factoring,  (x^  -  4)  (x-  +  3)  =  0. 

Putting  x^  —  4:  =  0,  or  x~  =  4,  we  liave  x  =  ±2. 

Putting       a;2  +  3  =  0,  or  x'^  =  —  3,  we  have  x  =±  V—  3. 
Checking  x  =2,    2*  -  22  -  12  =  16  -  4-  12  =  0. 
Checking  a:  =  V^3,   (V3)4(V^1)4-  (\/3)2(V^^)2-  12 

=  9(+  1)- 3(- 1)  -  12  =  12  -  12  =  0. 
Let  the  student  check  for  x  =  —  2  and  x  =  —  V—  3. 

Example  2.     Solve  (a;^  +  2)2-  7(a;2  +  2)  +  12  =  0. 

Consider  x^  +  2  as  the  unknown  and  call  it  z. 

Then  the  equation  is  z'^  —  1  z  -\-  12  =0. 

Factoring,  (0  —  4)  (s  —  3)  =  0. 

From  2;  —  4  =  0  or  x2  +  2  —  4  =  0, 

we  get  x'^  =  2  and  x  =±  V'2. 

From  2-3  =  0  or  x2 +  2-3  =  0, 

we  get  a;2  =  1  or  X  =  ±  1. 

Check.     Substitute  in  the  original  equation, 

WRITTEN  EXERCISES 

1.  x*-5x''-h(j  =  0.  4:.    2x*-x'-6  =  0. 

2.  x'  +  5x''-24:  =  0.  5.   ox'-10x^-^S  =  0. 

3.  x^- 3^2 -70  =  0.  6.   a;^- 7x2 +  12  =  0. 

7.  (a;2-5)2-9(a;2_5) +  20  =  0. 

8.  (x2  -\-x-  2)2  +  3  (x'  4-  X  _  2)  -  10  =  0. 

9.  (x2  _  5  .7;  +  4)2  -  (x2  -  5  X  +  4)  -  20  =  0. 
10.  (x2_3a;.-4)2  +  7(.r2-3x-4)-8  =  0. 

*  In  case  Articles  20(>-200  have  been  omitted,  then  the  solutions  which  in- 
volve the  use  of  imaginary  numbers  may  be  omitted  here. 


290  QUADRATIC   EQUATIONS 

Equations  Containing  Radicals. 

Example  3.     Solve  x  —  5  -j-  2\/x  —  5  =  8. 


Regard  Vx  —  5  as  the  unknown  and  call  it  z.     Then  a:  —  5  is  the 
square  of  z. 

Then  z"^  =  (VT^^y  =  x  -  6. 

Hence  the  equation  is  z-  +  2  z  —  8  =  0. 

Factoring,  •  (z  +  4)(z  -  2)  =  0. 

Heuce  z  =—4  and  z  =2. 


Then  Vcc  —  5  =  —  4  and  Vx  —  5  =  2. 

Hence  a:  —  5  =  16  and  x  —  5  =  4. 

From  which  a;  =  21  and  x  =  9. 

Note  that  x  =  2\  does  not  satisfy  the  original  equation,  but  that  x  =  9 
does. 

Example  4.     Solve  V^c  —  5-\/x  +  6  =  0. 

Call  Vx  the  unknown  and  represent  it  by  z.     Then  the  first  term,  Vx, 
is  the  square  of  z. 

Then  z^  =  {Vx^  =  (x^  y  =  x^  =  Vx. 

Hence  the  equation  is  z''^  —  5  z  -\-  6  =  0. 

Factoring,  (0  —  S)(z  —  2)=  0. 

Hence  z  =Z  and  z  =  2, 

from  which  •  Vx  =  3  and  Vx  =  2. 

Hence  a:  =  81  and  x  =  16. 

Check.     Substitute  each  value  of  x  in  the  given  equation. 

WRITTEN  EXERCISES 

Find  all  the  roots  and  check  in  the  iirst  eight : 

1.  ic  +  TVo- -30  =  0.'  6.    Va;- 3-</a5-10  =  0. 

2.  ■Vx  +  3Vx-2S  =  0.         6.    .r  +  3-4\/.M^  +  3  =  0. 


3.    a;-4Va;-12  =  0.  7.    x   -2  -  6Vx -2 -7  =  0. 


4.   a;  +  5  Vic  —  14  =  0.  8.   a;  +  5  —  V.i-  -h  5  =  6. 


9.    a;2  +  3  .T  +  2  +  3  V.r2  +  3  .r  +  2  =  0. 
10.    x^  —  7  X -\- 10 -\-  Vx^'^^7~x^l{)  =  0. 


ll.*2  a.-2  +  3  .r  +  6  -  2  V2a;2  +  3a;  +  9  =  0. 


PROBLEMS   INVOLVING   QUADRATICS  291 

HISTORICAL  NOTES 

Quadratic  Equations,  The  solution  of  the  quadratic  was  effected  by 
the  Greek  matlieiiiatician  Diophantus  in  much  the  same  manner  that  we 
solve  it.     He  did  not,  however,  admit  negative  roots. 

Alkarismi,  an  Arabian,  divided  all  quadratic  equations  into  five  classes ; 
namely,  ax-  =  bx,  ax'^  =  c,  ax-  +  bx  =  c,  ax-  -\-  c  =  bx,  ax^  =  bx  +  c  when 
rt,  6,  c  are  all  positive  numbers.  This  is  an  interesting  instance  of  the 
complexities  which  we  encounter  when  we  fail  to  recognize  that  letters 
may  represent  negative  as  well  as  positive  numbers.  Alkarismi  did  not 
admit  negative  roots,  but  he  recognized  the  possibility  of  a  quadratic  equa- 
tion having  two  positive  roots.  Bhaskara,  a  Hindu  mathematician,  gives 
X  =  50  and  x  =  —  5  as  roots  of  x-  —  45  x  =  250  but  remarks  that  ' '  The 
second  root  is  not  to  be  taken,  for  it  is  inadequate;  people  do  not  approve 
of  negative  roots."  It  was  only  after  the  time  of  Descartes  (see  page  240) 
that  negative  roots  came  into  perfectly  good  standing. 

Imaginary  Roots.*  The  imaginary  roots  of  equations  were  entirely  re- 
jected by  the  Ancients.  Cardan  in  Italy  (1501-1576),  and  Harri<n  in 
England  (1560-1621),  mentioned  them,  but  simply  said  that  they  indi- 
cated that  the  equations  which  gave  rise  to  them  represented  impossible 
conditions.  It  was  not  until  the  first  decade  of  the  nineteenth  century 
when  Argand  in  Switzerland  and  Gauss  in  Germany  (see  next  page)  gave 
a  geometrical  interpretation  of  imaginaries  that  they  came  to  be  fully 
understood.  In  this  respect  imaginaries  shared  the  fate  of  negative  num- 
bers, which  were  not  fully  understood  until  Descartes  in  France  applied 
them  to  geometry. 

PROBLEMS  INVOLVING   QUADRATICS 

Find  two  solutions  in  each  case,  and  determine  whether  both 
are  applicable  to  the  problem. 

1.  Find  two  consecutive  integers  whose  product  is  272. 

2.  Find  two  consecutive  integers  whose  product  is  182. 

3.  Find  two  consecutive  integers  whose  product  is  992. 

4.  The  sum  of  the  squares  of  two  consecutive  integers  is 
1-45.     Find  the  numbers. 

5.  Find  a  number  such  that  four  times  the  number  added  to 
12  is  equal  to  the  square  of  the  number. 


292 


QUADRATIC   EQUATIONS 


6.  Find  a  number  such  that  nine  times  the  number  is  equal 
to  its  square  plus  18. 

7.  A  picture  is  12  inches  wide  and  17  inches  long  inside 
the  frame.  How  wide  is  the  frame  if  its  diagonal  is  25 
inches  ? 

8.  A  rectangle  is  12  inches  wide  and  16  inches  long.  How 
much  must  be  added  to  the  length  to  increase  the  diagonal  by  4 
inches  ? 


^^ 

2 

X ^ 

^-                                                                   ~~-^ 

1 ~        ~    —     " 

1 

X 

^^ 

j/^       ^^ 

>/-'' 

y/^  ^^"                    12 

1 

^                    16 

Let  X  =  number  of  inches  to  be  added  to  the  length.  The  diagonal 
of  the  original  rectangle  is  Vl22  +  16"^  =  20.  Hence  the  diagonal  of  the 
required  rectangle  is  24. 


Then 


or 


Solving, 


and 


12-^  +  (16  +  a:)2  =  242, 

x2  +  32a;-  170  =  0. 
a:i=._16  +  12v'3  =  4.78, 
X2  =  -\6  -12\/S=-  .36.78. 

The  negative  solution  obtained  here  may  be  taken  to  mean  that  if  the 
rectangle  is  extended  in  the  opposite  direction  from  the  fixed  corner,  we 
shall  get  a  rectangle  which  has  the  re<iuired  diagonal.     See  the  figure. 

9.    How  much  must  the  width  of  the  rectangle  in  Problem 
8  be  extended  so  as  to  increase  the  diagonal  by  4  inches? 

10.  A  rectangle  is  21  inches  long  and  20  inches  wide.  The 
length  of  the  rectangle  is  decreased  twice  as  much  as  the  width, 
thereby  decreasing  the  length  of  the  diagonal  4  inches.  Find 
the  dimensions  of  the  new  rectangle. 


Karl  Friedrich  Gauss  was  born  at  Brunswick,  Germany,  in 
1777  and  died  at  Gottingen  in  1855.  Justly  called  the  greatest 
mathennatician  of  modern  times,  he  was  scarcely  less  famous 
as  physicist  and  astronomer ;  in  fact,  he  was  director  of  the 
observatory  and  professor  of  astronomy  at  Gottingen  during  the 
last  forty  years  of  his  life. 

He  wrote  on  almost  every  phase  of  mathematics,  and  in  many 
subjects  the  modern  development  is  largely  due  to  his  genius. 

Gauss  was  the  first  to  explain  satisfactorily  the  nature  of  imagi- 
nary quantities  and  to  put  their  use  in  solving  equations  on  a 
systematic  and  scientific  basis.  He  brought  into  general  use  the 
symbol  /for  the  imaginary  unit  V—  1. 


PROBLEMS   INVOLVING   QUADRATICS 


293 


11.  The  distance  in  feet  which  a  body  falls  in  t  seconds  is 
expressed  by  d  =  vt-{- 16  i^,  where  v  is  the  velocity  with  which 
the  body  starts. 

How  long  will  it  take  a  body  to  fall  from  the  top  of  the 
Eiffel  tower,  a  distance  of  900  feet,  if  it  starts  with  a  velocity 
of  100  feet  per  second  ? 

Suggestion.     Solve  the  equation 

16  «2  +  100  i  =  900. 

12.  How  long  will  it  take  a  body  to  fall  900  feet  if  it  starts 
with  a  velocity  of  50  feet  per  second  ? 

13.  How  long  will  it  take  a  body  to  fall  900  feet  if  it  starts 
at  10  feet  per  second  ? 

14.  How  long  will  it  take  a  boinb  to  fall  from  an  aeroplane 
900  feet  high  if  it  starts  with  no  initial  velocity  ? 

15.  A  and  B  start  from  a  certain  cross  roads  at  the  same 
time,  A  going  north  4  miles  per  hour  and  B  going  east  3  miles 
per  hour.  In  how  many  hours  will  they  be  16  miles  apart, 
measuring  in  a  straight  line  across  country  ? 

Solution.     Let  t  equal  the  required  number  of  hours. 
Then       (4  t)'^  +  (3  0"  =  16-^  =  256. 
16  t~  +  9  f^  =  256. 
25  «2  =  256. 
5f=±16.  ^^ 

t=±H. 

The  solution  «  —  —  3^  may  be  interpreted  as  meaning 
that  if  the  two  men  had  been  traveling  along  these  roads 
before   reaching   the   cross  road,  they  would  have  been  16  miles  apart 
3^  hours  before  meeting. 

16.  In  the  preceding  problem  if  A  goes  5  miles  per  hour  and 
B  goes  4  miles  per  hour,  in  how  many  hours  will  they  be  24 
miles  apart  ?     Ans.  3J  hours. 


294  QUADRATIC    EQUATIONS 

17.  The  sides  of  a  right  triangle  are  6  and  8  inches  respec- 
tively. How  much  Hiiist  be  added  to  each  side  in  order  to 
make  the  hypotenuse  20  inches  ?  It  is  understood  that  each 
side  is  increased  by  the  same  amount. 

18.  By  what  number  must  each  side  in  Example  17  be  multi- 
plied in  order  to  make  the  hypotenuse  20  ? 

19.  A  park  is  120  rods  long  and  80  rods  wide.  It  is  decided 
to  double  the  area  of  the  park,  still  keeping  it  rectangular,  by 
adding  strips  of  equal  width  to  one  end  and  one  side.  Find 
the  width  of  the  strips. 

20.  A  fancy  quilt  is  72  inches  long  and  56  inches  wide.  It 
is  decided  to  increase  its  area  10  square  feet  by  adding  a  border. 
Find  the  width  of  the  border. 

21.  A  farmer  starts  cutting  grain  around  a  field  120  rods 
long  and  70  rods  wide.  How  wide  a  strip  must  be  cut  to  make 
12  acres  ? 

22.  In  a  rectangular  table  cover  24  by  30  inches  there  are 
two  strips  of  drawn  work  of  equal  width 
running  at  right  angles  through  the  center 
of  the  piece.  What  is  the  width  of  these 
strips  if  the  drawn  work  covers  one  tenth 
of  the  whole  piece  ? 

23.  An  athletic  field  is  800  feet,  long 
and  600  feet  wide.  The  field  is  to  be  extended  by  the  same 
amount  in  length  and  width  so  that  the  longest  possible  straight 
course  (the  diagonal)  shall  be  1100  feet.  How  much  must  the 
field  be  extended  in  each  direction  ?     Ans.  71.36  feet. 

24.  A  farm  is  320  rods  long  and  280  rods  wide.  There  is  a 
road  running  around  the  boundary  of  the  farm  and  lying 
entirely  within  it.  Find  the  width  of  the  road  if  the  area 
covered  by  it  is  2384  square  rods. 


I —  J- 


1 

G 

X-12 

._. 

PROBLEMS  INVOLVING  QUADRATICS  295 

25.  A  rectangular  park  is  480  rods  long  and  360  rods  wide. 
A  walk  is  laid  out  completely  around  the  park,  and  a  drive  is 
laid  out  through  the  length  of  the  park  parallel  to  the  sides. 
What  is  the  width  of  the  walk  if  the  drive  is  3  times  as  wide 
as  the  walk  and  the  combined  area  of  the  walk  and  the  drive 
is  3110  square  rods  ? 

26.  A  square  piece  of  tin  is  made  into  an  open 
box,  containing  $64  cubic  inches,  by  cutting  out 
a  6-inch  square  from  each  corner  of  the  tin  and 
then  turning  up  the  sides.  Find  the  dimensions 
of  the  original  piece  of  tin. 

27.  A  rectangular  piece  of  tin  is  8  inches  longer  than  it  is 
wide.  By  cutting  out  a  7-inch  square  from  each  corner  and 
turning  up  the  sides,  an  open  box  containing  1260  cubic  inches 
is  formed.     Find  the  dimensions  of  the  original  piece  of  tin. 

28.  By  cutting  out  a  square  8  inches  on  a  side  from  each 
corner  of  a  sheet  of  metal  and  turning  up  the  sides,  we  obtain 
an  open  box  such  that  the  area  of  the  sides  and  ends  is  4  times 
the  area  of  the  bottom.  Find  the  dimensions  of  the  original 
sheet  if  it  is  twice  as  long  as  it  is  wide. 

Ans.  41.48  in.  by  20.74  in. 

29.  An  open  box  whose  bottom  is  a  square  has  a  lateral 
area  which  is  400  square  inches  more  than  the  area  of  the 
bottom.  Find  the  other  dimensions  of  the  box  if  it  is  10 
inches  high.  (By  lateral  area  is  meant  the  sum  of  the  areas 
of  the  four  sides.) 

30.  A  box  whose  bottom  is  4  times  as  long  as  it  is  wide 
has  a  lateral  area  600  square  inches  less  than  4  times  the  area 
of  the  bottom.  Find  the  dimensions  of  the  bottom  if  the  box 
is  6  inches  high. 


296  QUADRATIC   EQUATIONS 

REVIEW  QUESTIONS 

1.  Explain  the  method  of  solving  a  quadratic  equation  by 
factoring.     Can  you  apply  this  to  (x  -\-  V){x  —  2)  =  5  ?     Why  ? 

2.  Explain  the  method  by  completing  the  square.  How  can 
this  be  done  so  as  to  avoid  fractions  until  the  end  of  the  solu- 
tion ? 

3.  Explain  the  solution  of  the  quadratic  by  means  of  the 
formula.  Why  does  solving  the  equation  ax"^  -{-  bx  -\-  c  =  0 
give  a  formula  for  all  quadratic  equations  ?  Describe  this  type 
form  of  quadratic  equation  in  words. 

4.*  How  many  roots  has  a  quadratic  equation  ?  Find  the 
roots  of  a;^  4-  2  a;  +  5  =  0.  What  are  these  roots  called  ?  Solve 
ic^  -f  2  a;  —  5  =  0.     What  are  these  roots  called  ? 

5.  How  are  the  roots  of  the  equation  x^  -{-  px  -^  q  =  0  related 
to p  and  q?     How  may  this  be  used  in  checking  the  solutions  ? 

Use  2  ic^  —  4  ic  +  1  =  0  to  show  how  the  sum  of  the  roots  and 
product  of  the  roots  may  be  found  from  the  given  equation. 

6.  When  is  an  equation  said  to  be  in  the  quadratic  form  ? 
Which  of  the  following  are  in  this  form  ? 

ic6  _  3  ^^  _  4  ^  0  . 

a;2  _  4  _  2  VaJ23^  -  3  =  0 ; 
(a;2  +  1 )  2  +  3  (^2  +  1 )  +  2  =  0 ; 

ic2  -  3  a?  + 1  +  Va;2  _3a;4-4  +  5  =  0. 

How  can  the  last  one  be  put  into  the  quadratic  form  ? 

7.  Do  both  roots  of  a  quadratic  equation  necessarily  satisfy 
the  conditions  of  the  problem  from  which  such  an  equation 
may  be  derived  ?  In  checking  the  solution  of  a  problem  is 
it  sufficient  to  make  the  test  alone  in  the  equation  derived 
from  the  problem  ? 


CHAPTER    XIX 

SYSTEMS   OF   QUADRATICS 

ONE   QUADRATIC  AND   ONE   LINEAR  EQUATION 

211.  When  two  simultaneous  equations  are  given,  one  quad- 
ratic and  one  linear,  they  may  be  solved  by  the  process  of  sub- 
stitution, which  was  used  (§  150)  in  the  case  of  two  linear 
equations. 

Solutions  by  Factoring. 

Example.     Solve  the  equations  : 

\^x^-f-  =  -16.  (1) 

\x-3y  =  -12.  (2) 

From  (2)hj  S,  x  =  Sy  -  12.         (3) 

Substituting  (3)  in  (1),  (Sy  -  12)^  -  y'^=-  16.  (4) 

From  (4)  by  F,  9 1/^  -  72  ^  +  144  -  y-  =  -  16.  (5) 

From  (5)  by  F,  A,  Sy^-12y  -\-  160  =  0.  (6) 

ByZ>,  y^-9y-\- 20  =  0.  (7) 

Factoring,  (y  —  5)(y  —  4)=  0.  (8) 

Hence,  2/  =  5,  and  y  =  4:.  (9) 

Substitute  y  =  5  in  (3)  and  find  x  =  S. 

Substitute  ?/  =  4  in  (3)  and  find  x  =  0. 

Therefore  (1)  and  (3)  are  satisfied  by  the  two  pairs  of  values, 

X  =  3,  ?/  =  5  ;         and  x  =  0,  y  =  4:. 
Check  by  substituting  these  pairs  of  values  in  (1)  and  (2). 

Whenever  the  quadratic  in  one  unknown,  resulting  from  the 
substitution,  can  be  solved  by  factoring,  this  method  should  be 
used  as  in  the  above  solution,  beginning  at  equation  (8). 

When  the  solution  by  factoring  is  not  possible,  then  the 
formula  should  be  used,  as  in  the  example  on  page  299. 

297 


298 


SYSTEMS   OF   QUADRATICS 


EXERCISES 

In  the  maimer  just  illustrated  solve 
|5a;2-f  12?/2=128. 

x"  +  1^  =  1. 

:2x-y  =  6, 
[4.x''  +  o2f  =  36. 

X  -\-3y  =  6, 

a;2  +  32/'=12. 

x-2y=-2, 

x"^  —  6y^  =  10. 
,  X  -  16  2/  =  -  120, 
j  7  0)2  +  2  2/2  =  585. 

7x  +  9y  =  SS, 

Ta.-2  +  9  2/-  =  736. 


2. 


3. 


4. 


5. 


8. 


9. 


10. 


11. 


12. 


13. 


14. 


\x-y  =  6, 

I  a;2  —  7  2/2  =  36. 
3x  +  2y  =  7, 
Sx^^Sy^  =  S5. 
x-Sy  =  -ll, 

1 3  a;2- 16  2/2  =  11. 

lx-y  =  -T, 

[4  a;2  4-32/2  =  147. 

|.«-2/  =  2, 

I  a;2  —  5  2/^  =  4. 

ix-y  =  l, 

[3a:2-22/'  =  -5. 

f5a;_7,/^_28, 

|l5a;2  +  492y2  =  784. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 
23. 
24. 
25. 
26. 
27. 
28. 


the  following : 

f6.T-7?/  =  18, 
1 36.^2 -7  2/- =  324. 
lx-9y  =  2, 
|a;2_45  2/2  =  4. 
{x-\-y  =  S, 
|l3x2  + 3  2/2  =  160. 
i2x-5y  =  -16. 
[4  0:2  +  15  2/- =  256. 
7x-\--iy  =7, 
^49^2-8  2/2  =  49. 
[a; -3?/ =-12, 
|a;2-2/'  =  -16. 
|a;H-22/  =  3, 
I  3  x'  +  1 2/'  =  "• 
f  3  .^•  -  2/  =  5, 
|3a;2-2/2=ll. 

Lt  +  22/=7, 
4 .1-2  _  ?/2  ^  32. 

5a;  +  22/  =  13, 

^x-{-y-  =  17. 
I  4  .y  -  2/  =  3, 
ia;2  +  22/2  =  54. 
I  a;  =  2/  -  1, 
[5x2  +  227  =  53. 
j22/  =  3.r  +  l, 
[3x2  =  2/2  +  2. 
[x  =  32/+3, 
2x2-52/2  =  67. 


SOLUTIONS  BY  FORMULA 


299 


Example.     Solve 


212.    Solutions  by  the  Quadratic  Formula. 

a:  +  v/  =  3,  (1) 

3  a;2 -2/2  =  14.  (2) 

From  (1),  y  =  ?j-x.  (3) 

Substituting  in  (2)  and  reducing, 

2x2 +6x- 23  =  0.  (4) 

Equation  (4)  is  in  the  form  ax^  +  6x  +  c,  in  which  a  =  2,  6  =  6,  c  =  —  23. 
Substituting  in  the  formula,  §  205, 


^  -  6  J:  \/30  -  4  ■  2(-  23)  ^  -  3  jr  V55 
^~  4  2 


(5) 


Hence  Xi  =  2.21  and  X2=—  5.21. 

Substituting  these  values  of  x  in  (1)  we  have  as  the  approximate  roots, 

ari  =  2.2n  a;o=-5.21 

\    and 
yi  =  0.79  J  2/2  =  8.21 

t/i  and  ^2  are  here  used  to  designate  those  values  of  y  which  correspond 
to  X\  and  X2,  respectively. 

EXERCISES 

In  the  above  manner  solve  the  following  systems  of  equa- 
tions, finding  in  each  case  two  pairs  of  values. 


t  x^  +  ?/2  =  13. 
x  +  y  =  9, 
x'~  +  7/2  =  41. 

x-{-y  =  13, 
xy  =  42. 

(  3  X  —  y  =!=  5y 


7. 


6. 


[x'~  +  f  =  25. 

.94 

,  X  -  ?/  =  -4. 
x-y  =  l, 
a;2       ?/2  _  1 
36     i6~2' 


10. 


11. 


12. 


13. 


2x-^y  =  5, 

3x^-of-  =  l. 
(  X  —  y  =  o, 
|aj2-32/-  =  13. 
(3x-4:y  =  l, 
I  a;2  -  2/2  =  24. 

2x'-3xy  +  y'  =  S. 
^x-y  =  l, 
\4:x''-\-2xy-y^  =  19. 

5x  +  y  =  12, 

2x^^-3xy-\-f-  =  0. 
(x  +  y  =  9, 
|.t2-22/2  =  -7. 


300  SYSTEMS   OF   QUADRATICS 

213.    Special  Case.     Certain  systems  involving  quadratics  may- 
be solved  by  special  devices. 

x  +  y  =  5,  (1) 


Example  1.     Solve  l^.  +  ^.^i3.  (2) 

Solution.     Squaring  (1)  x-  -{- 2  xy  -\- y'^  -  25  (3) 

Subtracting  (2)  from  (3),  ^ +  y^  =  13 

2xy  =12  (4) 

Subtracting  (4)  from  (2),  3.2  -j- y^  =  13 

2r/^  —  2xy-\-y^  =  l 

Extracting  the  square  root,  x  —y  =  ±  I  (4) 

fic  +  w  =  5  ^    ,    [05  =  3 

Solvmg  the  system  i     _  "^  _         we  hnd   j     _ 

I/yt  _i    nM  rc  \  or  ^z  2 

_     __  .  we  find  3 

^       ^  ^         f         xy  =  S.  (1) 

Example  2.     Solve    \^2_^y2^20.  (2) 

Adding  twice  equation  (1)  to  (2),  x"^ -\- 2  xy  +  y'^  =  36.  (3) 

Taking  square  roots,  x  +  y  =±6.  (4) 

Subtracting  twice  equation  (1)  from  (2),  x~  —  2  xy  +  y^  =  4.  (6) 

Taking  square  roots,  x  —  y  =  ±2.  (6) 

Solving  the  systems 
ix  +  y  =  e         (x  +  y  =  Q    .        \x  +  y=-6,        \x  +  y=-6 
\x-y  =  2'        \x-y=-2'        \x-y  =  2      '        \x-y=-2' 

we  find  four  pairs  of  solutions. 

WRITTEN  EXERCISES 

1.    Solve  by  the  above  method  Examples  1,  2,  3,  in  the  pre- 
ceding exercise. 

Solve  by  the  same  method : 

[x-y  =  l.  [x^-  ^y  +  2/  =  19- 

xy  =  12,  g      I  xy  =  -  6, 

a;2 +2/2  =  25.  '     [x'^-Sxy -{-y'  =  31. 

xy  =  6,  ^      (  ^!/  =  -t, 

x^-xy-{-y^  =  7.  '    \2x''-{-5xy -\-2y^=5'i. 


SYSTEMS   OF   QUADRATICS  301 

A   SYSTEM   OF   TWO    QUADRATICS* 
214.    Homogeneous  Quadratics.     A  quadratic  equation  in  x  and 
y  is  homogeneous  when  every  term  is  of  the  second  degree  in 
X  and  y,  that  is,  when  every  term  contains  a^,  xy,  or  y-. 

E.g.     2  a:2  -I-  3  jcy  -I-  y2  —  0  is  homogeneous  in  x  and  y. 

a;2  _|_  3  a;  ^  y2  —  0  is  not  homogeneous,  because  the  term  3  ic  is  of  the 
first  degree  ;  and  x^^  xy  -\-  y^  =  5  is  not  homogeneous  because  the  term  5 
contains  neither  x  nor  y. 

If  a  quadratic  equation  which  is  homogeneous  in  x  and  y  is 

divided  through  by  o;^,  every  term  will  then  contain  -,  and  the 

X 

equation  may  be  solved  for  ^  considered  as  the  unknown,  as  in 
the  following  example : 

Example.     Solve  for  ^  the  equation  2x^  -\-'^xy  -\-y'^  =^^. 


X 

Dividing  by  x^,  and  writing  ^^  first,  we  have 


t 

x^ 


^  +  3^+2=0. 
x^         X 

This  is  now  a  quadratic  in  which  ^  is  the  unknown. 

X 

Factoring,  f^-  +  2  V^  +  l")  =  0. 

Hence,  ^  =  —  2  and  ^  =  —1. 

X  X 

Hence   the  homogeneous  quadratic  equation,  2  x-  -\-  S  xy  +  y'^  =  0,  is 
equivalent  to  the  two  linear  equations,  y  ——  2x  and  y  =—  x. 

WRITTEN  EXERCISES 

Solve  the  following  equations,  regarding  ^  as  the  unknown. 

X 

1.  y^-3xy-\- 2x^=0.  6.  a^ -hSxy -[-16y'^  =  0. 

2.  y^-\-7xy-S0x''  =  0.  7.  2x^  +  3xy -2  y^  =  0. 

3.  2/2  -  11  a;?/ +  30  ar^  =  0.  8.  x^ -{-2xy -63y^  =  0. 

4.  y^-7xy-lSx''  =  0.  9.  2  .r^  -  14  x?/ -  60  y^  =  0. 

5.  3x''-^llxy-20y^  =  0.  10.  24:0^ -12xy -12y^  =  0.  ■ 

*  Articles  214-218  may  be  omitted  without  destroying  the  continuity. 


302  SYSTEMS   OF   QUADRATICS 

215.    Systems  in  which  One  Equation  is  Homogeneous.     If  one 

of   two  quadratic  equations  in   x  and.  y  is  homogeneous,  the 
complete  solution  may  be  found  as  in  the  following: 

Example.     Solve       {  ^  ^:  +  f  ^^ +J:  = '"'  W 

Since  (1)  is  homogeneous,  we  may  solve  it  for  ^  and  find,  as  in  the 
example  solved  in  §  214,  ^ 

^=-2  and  ^=-1. 

X  X 

Hence,  y  =—  2x  and  y  =—  x. 

First.     Substituting  ?/  =  —  2  x  in  (2) ,     x^  —  5  a;  +  8  a;2  -  4.                (3) 

Transposing,  9  x^  _  5  ic  —  4  =  0. 

Factoring,  (a;  —  1 )  ( 9  x  +  4)  =  0. 

Hence,  x  =  1  and  x  =  —  |. 
Hence  vs^e  have 

|x  =  l,  \x=-^, 

\y=-'2x  =  -2,  ^"^     |2/=-2:>:=-2(-|)=f. 

Second.     Substituting  ?/=— x  in  (2),     x-  — 5x  +  2x2  =  4.  (4) 

Transposing,  ^x-— ox  —  4^  =  0. 

This  is  in  the  form  ax^  +  6x  +  c  =  0,  in  which  a  =  3,  &  =  —  5,  c  =—  4. 


Solving  by  the  formula,  §  205,         x  =  ^  + /"^'^  and  x  =  ^  ~  ^ ' ^ 
Hence  we  have 


6    .  6 


^  _  5+V73                                   f     _  5  -  V73 
JO  —  .  i  X  — 1 

6        '  6 


y  =  -.  =  :z±^pm. 


-5-f-\/73 

y  =—  x  = ■ 

^  6 


216.  The  solutions  must  be  carefully  collected  in  pairs.  For  in- 
stance, when  y=  —2  xis  substituted  in  equation  (2)  above  we  get 
equation  (3),  from  whicli  x=l  and  x  =  —  -J.     But  when  y  =  —  x 

is  substituted  in  (2)  we  get  (4),  from  which  x  =  — ^ ^    and 

5-V73  ^ 


X  = 


G 

Hence  y  =  —  2x  goes  only  with  x  =  l  and  it'=  — J;  while 

7       -.u          5+V73       1         5-V73 
y  =  —  X  goes  onty  with  a;  =  — ' and  x  = — . 


A   SYSTEM  OF  TWO    QUADRATICS  303 

WRITTEN  EXERCISES 

Solve  the  following  systems  and  collect   the   solutions   in 
pairs. 

^      (Sx''  +  xi/  =  S5,  ^  (12x'-\-xy-i/  =  S, 

|5/4-8a;?/-4x2  =  0.  "  [^Ox^ +  7  xij -3i/  =  0. 

\3y^-13xy-\-12x'  =  0.  '    \  2xy -{-6x- 5y  =  0. 

3      (y^  +  ^y—^=^,  Q     (  x"^ -{■  4:  xy -\- 5  y^  =  36, 


1 12 a.'2  -xy  -67/-  =  0.  [x'' -^  xy —  2y^  =  0 


Example.     Solve 


217.    Systems  in  which  both  equations  are  homogeneous  except 
for  the  numerical  terras. 

x^-\-3xy-\-2f-=.15,  (1) 

4  a^  +  5  2/2  =  24.  (2) 

Solution.     The  plan  is  to  eliminate  the  numerical  terms,  as  follows : 
Multiplying  (1)  by  8     f  8  x^  +  24  a;?/  +  16  y'^  =  120.  (.3) 

aud  (2)  by  5,  1  20  ic2  +  25  y'^  =  120.  (4) 

Subtracting  (3)  from  (4),  12  x^  -  24  a:y  4-  9  y^  :=  o.  (o) 

Since  (5)  is  homogeneous,  3^  —  8^  +  4  =  0.  (6) 


Solving  (6)  for  ^,  2^  =  2  or  ^  =-, 

X  X  X      3 

from  which  y  =  2xovy  =  ^x. 

Substituting  ?/  =  2  5c  in  (2),  we  have  4 x^  +  20 x^  =  24. 

Hence,  x=±l. 

Substituting  y  =  ^xin  (2),  we  have  4:X--\-  -^^x^  =  24, 
from  which  x  =  f  VT  and  x=—  f  \/7. 

Collecting  results  in  pairs,  we  have 


x  =  l 
y  =  2' 


x=-l       fa;  =  fV21       (x  =  -iV2\ 
_y=-2'     i?/  =  f\/2l'     \y=--jV2i' 


WRITTEN  EXERCISES 

Solve  and  arrange  the  results  in  proper  pairs : 

^      {2xy-x^-  =  lD,  ^      |.,^  +  .^.^  =  _28, 

^      ix'-2xy  =  -12,  ^      jy^-^xy  =77, 

53/2  +  3.17  =  104.  '     [x2-x?/  =  -12. 


Example  1.     Solve 


304  SYSTEMS   OF   QUADRATICS 

218.  Other  Special  Ci^ses.  A  few  other  special  cases  are  illus- 
trated in  the  following  examples,  some  of  which  include  equa- 
tions of  higher  degree  than  the  second. 

^+2/^  =  9,  (1) 

x+y=Z.  •  (2) 

Cubing  (2),  x^  +  3  a;2y  +  3  ic?/^  +  ?/3  =  27.  (3) 

Subtracting  (1)  from  (3),       3  ofiy  +  3  xy'^  =  18.  (4) 

Dividing  by  3,  x^y  +  xy^  =  6.  (5) 

Multiplying  (2)  by  xy,  xhj  +  xy^  =  3  xy.  (6) 

Subtracting  (5)  from  (6),  Sxy  =  Q. 

xy  =  2.  (7) 

Solving  (7)  and  (2)  as  in  §  213,  we  get  f  ^  ==  ^'   and    |  ^  ^  ]•> 

[y  =  l,  12/ =  2. 

Another  method  is  to  divide  (1)  by  (2),  obtaining 

x"-^  —     xy  -\-  y'^  =  S 
Squaring  (2),  x:^  -\- 2  xy  -j-  y'^  =  9 


Subtracting,                                             Sxy  =6 

The  remaining  steps  are  the  same  as  above. 

Example  2.    Solve  |^' ~  2/"  =  ^^' 
[x^  —  y^  z=  5. 

(1) 

(2) 

Dividing  equation  (1)  by  (2),                 x-  +  y'^  =  13. 

(3) 

Solving  equations  (2)  and  (3),  we  find, 

{x  =  S  ,    {x=-3  . 
\y  =  2'    \y=-2' 

x  =  3 

jx 

[y 

=  — 

3, 

[2/=-2 

=  2. 

Examples.     Solve   /       /  ~^^  ""'i^l''' 

(1) 
(2) 

[2y^—xy-\-x^=Sx. 

Muhiplying  (1)  by  8,                       8  ?/2  +  16  xy  =  56  x. 

(3) 

Multiplying  (2)  by  7,           Uy"^  -  1  xy  +  7  x^  =  56  x. 

(4) 

Subtracting  (3)  from  (4),    Gy'^  -  2Sxy  -j-1  x"^  =  0. 

(5) 

Factoring,                             (Sy  -  x)i2y  -  7  x)  =  0. 

(6) 

Hence                                                 V  =7  and  y  =  -^. 

3                   2 

(7) 

f  a:  =  0,                      \x-- 

=  9, 

Solving  V  =-  with  (1),  we  find   \         ^           and 

3            ^   ^'                   \y=^=0,              \y-- 

_x  _ 
"3~ 

3. 

7x                                   f*  =  ^' 
Solving  y  =^^  with  (1),  we  find  ]         7  x     r.    ^^^^ 

X  = 

4 

TT' 

7  x_ 

_14. 

[  ^  ~ir~ 

y  = 

2 

"11* 

PROBLEMS   INVOLVING   SYSTEMS   OF   QUADRATICS      305 
WRITTEN  EXERCISES 

Solve  the  following  and  arrange  the  results  in  pairs : 
^^     U-^ +7/3  =  243,  ^      (x'-y*  =  63, 

[^  X  -\-  y  =  9.  \oc^  —  y^  =  S. 

2  (x'-y'  =  15,  g      (2x'-\-4.xy-5y  =  0, 
\x'^  -{-  y'^  =  o.  [2  x^—  xy  +  y^—2  y  =  0. 

3  (^a^-f  =  12,  ^     ^x'-f  =  5^, 
'     IX  -y  =2.  '     ix-y=6. 

^      (x^-f  =  26,  ^      ix^-^f  =  2^3, 

\  X  —  y  =  2.  \  x^y  -\-  xy"^  —  162. 

Suggestion  for  Example  8.    Multiply  the  second  equation  by  3  and  add 
to  the  first,  thus  obtaining  a  perfect  cube. 

PROBLEMS   INVOLVING   SYSTEMS  OF   QUADRATICS 
In  each  case  find  all  the  solutions  and  determine  whether  all 
are  applicable  to  the  problem. 

1.  Find  two  numbers  whose  sum  is  25  and  whose  product 
is  156. 

2.  The  sum  of  two  numbers  is  35  and  their  product  is  300. 
Find  the  numbers. 

3.  The  sum  of  two  numbers  is  —  15  and  their  product  is 
—  700.     Find  the  numbers. 

4.  The  difference  of  two  numbers  is  13  and  their  product  is 
510.     Find  the  numbers. 

5.  The  sum  of  two  numbers  is  18  and  the  sum  of  their 
reciprocals  is  \.     Find  the  numbers. 

6.  The  difference  of  two  numbers  is  6  and  the  sum  of  their 
reciprocals  is  -j^g.     Find  the  numbers. 

7.  A  rectangular  field  is  20  rods  longer  than  it  is  wide. 
Find  its  dimensions  if  its  area  is  2400  square  rods. 

8.  A  rectangular  field  is  20  rods  longer  than  it  is  wide. 
Find  its  dimensions  if  its  area  is  50  acres. 


306  SYSTEMS   OF   QUADRATICS 

9.    The  sum  of  the  sides  of  a  right  triangle  is  14  and  the 
length  of  the  hypotenuse  is  10.     Find  the  length  of  each  side. 

10.  The  length  of  a  fence  around  a  rectangular  athletic  field 
is  1400  feet,  and  the  longest  straight  track  possible  on  the  field 
is  500  feet.     Find  the  dimensions  of  the  field. 

Suggestion.     Using  100  feet  for  the  unit  of  measure,  the  equations  are 

cc2  +  ij2  =  25. 

11.  The  difference  between  the  sides  of  a  right  triangle  is  8 
and  the  hypotenuse  is  40.     Find  the  lengths  of  the  sides. 

12.  A  room  is  5  feet  longer  than  it  is  wide,  and  the  distance 
between  two  opposite  corners  is  25  feet.  Find  the  length  and 
width  of  the  room. 

13.  One  side  of  a  right  triangle  is  8  feet,  and  the  hypotenuse 
is  2  feet  more  than  twice  the  other  side.  Find  the  length  of 
the  hypotenuse  and  of  the  remaining  side. 

14.  The  area  of  a  window  is  2016  square  inches  and  the 
perimeter  of  the  frame  is  180  inches.  Find  the  dimensions  of 
the  window. 

15.  The  area  of  a  rectangular  city  block,  including  the  side- 
walk, is  19,200  square  yards.  The  length  of  the  sidewalk 
around  the  block  wlien  measured  on  the  side  next  the  street  is 
560  yards.     Find  ilio  dimensions  of  the  block. 

16.  A  farmer  starts  to  plow  around  a  rectangular  field  which 
contains  48  acres.  The  length  of  the  first  furrow  around  the 
field  is  376  rods.     Find  the  dimensions  of  the  field. 

17.  A  rectangular  blackboard  contains  38  square  feet  and 
its  perimeter  is  27  feet.     Find  the  dimensions  of  the  board. 

18.  The  sum  of  the  squares  of  two  numbers  plus  five  times 
their  product  is  equal  to  445;  and  the  sum  of  their  squares 
minus  their  product  is  equal  to  67.     Find  the  numbers. 


REVIEW   QUESTIONS  307 

19.  The  diagonal  of  a  rectangular  mirror  inside  the  frame  is 
10  inches.  The  square  of  the  diagonal  of  the  frame  is  244 
inches.  Find  the  dimensions  of  the  mirror  if  the  frame  is 
2  inches  wide. 

20.  Find  the  altitude  of  a  right  triangle  whose  sides  are  6, 
8,  and  10,  taking  the  hypotenuse  10  as  the  base.  Also  find 
the  area  of  this  triangle. 

Suggestion.     Using  the  figure,  deduce  the 
following  equations : 

I  7-?  +  1/  =  30, 

\(10-x)'^  +  7j~  =  6-i.  10 

21.  Find  the  altitude  of  a  right  triangle  whose  sides  are  3,  4, 
and  5,  taking  the  hypotenuse  5  as  the  base. 

22.  Find  the  altitude  of  a  triangle  whose  sides  are  10,  8,  and 
14,  taking  the  side  14  as  the  base.     Also  find  the  area. 

23.  The  sum  of  the  squares  of  two  numbers  minus  four 
times  their  product  is  —  23 ;  and  the  sum  of  their  squares 
plus  three  times  their  product  is  61.     Find  the  numbers. 

REVIEW   QUESTIONS 
1.   Explain  the  method  of  solving  a  system  of  two  equations 
in   which  one    is    linear    and    one     quadratic.      How    many 
solutions  are  there  ?     Explain  how  to  solve  the  special  case 

J    „  ^  ~         ;  also  the  special  case  ]        '^V  —  ^ 
[x2  +  ^2^13    '  "^  lx+?/  =  10. 

2.*  When  is  a  quadratic  homogeneous  in  two  unknowns  ? 
How  can  a  system  of  two  quadratics  be  solved  when  at  least 
one  of  them  is  homogeneous  ?     How  many  solutions  are  there  ? 

3.=*  How  may  a  system  of  two  quadratics  be  solved  if  they 
are  both  homogeneous  except  for  the  numerical  terms  ? 

4.*  What  other  special  methods  of  solving  systems  of 
equations  were  used  in  the  exercises  on  p.  304  ? 


CHAPTER  XX 

THE   BINOMIAL  FORMULA* 

219.  We  have  already  considered  the  square  and  the  cube 
of  a  binomial,  namely, 

(a  +  hf  =  a'^  +  2ah  +  b^. 

(a  +  by  =  a^  +  Sa^b  +  3  ab^  +  ¥. 

If  we  multiply  (a  +  by  by  a  +  6,  we  have 

(a  +  6)4  =  a*  +  4  a^b  +  6  a^b'^  +  4  afts  +  &*. 

In  like  manner, 

(a  +  6)5  :zz  a5  +  5  ^^4^  +  lo  058^,2  +  10  a^^^  +  5  a6*  +  6^. 

The  right  members  of  these  equations  are  called  the  expansions 
of  the  left  members.  By  studying  these  expansions  we  see 
that  they  may  be  written  down  according  to  the  following  rule  : 

(1)  I/i  the  expansion  of  a  hinoinial  there  is  one  more 
term  than  the  number  of  units  in  the  index  of  the  power. 

(2)  The  first  letter  of  the  binomial  is  a  factor  of  all  terms 
in  the  expansion  except  the  last.  Its  exponent  in  the  first 
term  is  equal  to  the  exponent  of  the  binomial,  and  it 
decreases  by  one  in  each  succeeding  term. 

(3)  The  second  letter  is  a  factor  of  all  terms  in  the  ex- 
pansion except  the  first.  Its  exponent  in  the  second  term 
is  one,  and  in  each  succeeding  term  it  increases  by  one. 

(4)  Th"".  coefficient  of  the  first  term  is  one.  The  coeffi- 
cient of  the  second  term  is  the  same  as  the  index  of  the 
power.  The  coefficient  of  each  succeeding  term  is  found' 
by  multiplying  the  coefficient  of  the  preceding  term  by 
the  exponent  of  the  first  letter  in  that  term  and  dividing 
by  the  exponent  of  the  second  letter  increased  by  one. 

*  Articles  219-222  may  be  omitted  without  destroying  the  continuity. 

308 


THE   BINOMIAL   FORMULA  309 

220.   According  to  this  rule  we  may  write  the  expansions  on 
page  308  in  the  following  forms : 

(a+ &)2  =  a2_,_  2a6  4-2_J:52.  • 

(a  +  6)8  =  a=^+3  a26  +  —  a&2  ^  liljj  ^3. 
^  2  2.3 

{a  +  &)"  =  a*  +4a8?,  +  i_3  ^252  +i^3_2^^3  ^4^3^2^  ^4^ 

^  ^  2  2.3  2-3.4 

|5-4. 3-2.1^, 
2.3.4.5 

ORAL  EXERCISES 

1.  How  many  terms  are  there  in  the  expansion  of  (a  +  lif  ? 

2.  In  the  expansion  of  (a  +  Vf  is  there  any  term  in  which  a 
does  not  occur  ?     Is  there  any  term  in  which  h  does  not  occur  ? 

3     In  the  expansion  of  (a  4-  ISf  what  is  the  exponent  of  a  in 
the  first  term  ?  in  the  second  ?  in  the  third  ?  and  so  on. 

4.  In  the  expansion  of  (a  +  ISf  what  is  the  exponent  of  h  in 
the  second  term  ?  in  the  third?  in  the  fourth  ?  and  so  on. 

5.  In  the  expansion  of  (a  +  6)^  state  how  each  coefficient  is 
obtained. 

6.  Ask  and  answer  questions  similar  to  those  in  Exercises 
1  to  5  about  the  expansion  of  (a  +  hY  and  (a  +  hy. 

WRITTEN  EXERCISES 

Expand  each  of  the  following  by  the  binomial  formula : 
1.    {x-\-yf.  6.    {\-\-xf.  11.    (a-6)^ 


2.  {x-^y) 

3.  {x-^y) 

4.  (a;  +  l) 

5.  {x^-V)\ 


8.  (1  +  .t)^  13.    {x-V)\ 

9.  (a-6)^  14.    {\-x)\ 
10.    (a-6)^  15.    (1-a;)^ 


Suggestion  for  Examples  9-15  :  Write  a— 6  =  a+(—  6)  and  note  that 
even  powers  of  —  6  are  positive.^  while  odd  powers  are  negative. 


310  THE   BINOMIAL  FORMULA 

221.  Mathematical  Induction.  We  have  found  by  actual  mul- 
tiplication that  the  rule  of  §  219  holds  for  powers  up  to  the  fifth, 
and  we  wish  to  find  out  without  continuing  the  multiplication 
whether  it  holds  for  higher  powers.  This  is  done  by  the  process 
called  mathematical  induction,  which  is  as  follows : 

(1)  We  write  out  the  ^th  power  according  to  the  rule,  thus : 
(a  +  b)k  =ak  +  kai^'^b  +  M^UlD  ^^-252  +  H^  -  J )  (^  -  2 )  ^^-353  +^  etc. 

(2)  We  find  (a  +  6)^+1  by  multiplying  the  above  result  by  a  +  6  as 
follows : 

Multiplying  by  a, 

2  2.3 

Multiplying  by  &,         a«^6  +  A:a*-i&-' +  Mzil) 0^^-253  ^.  ... 

Adding  terms,   a*+i  +  (A;  +  l)a*&+ [^^^=^ +ifc'ja*-i&2  + 

/^(^:^1)(A>-21     ^^-l)\«;fc-263  .   ... 
V  2.3  2       J 

To  add  ^^^~  ^  +  k,  we  use  Principle  I,  thus 

k(k-l)      J      jfk-l      ^l^k(k  +  l)^{k  +  l)k_ 
2  L     2  J  2  2 

Sin^ilarly  ^(^  -  ^H^  -  ^)  +  Mi^  =  ^^^^^f^  +  l] 
2-3  2  2  L     o  J 

2L3J  2.3  2.3 

Hence,  we  have 
(a  +  &)*+!  =  a*+i  +(A;  +  l)a^b  +  ^^±ilMa*-i&2 

iJC^\){k){k-V)  ^,_,^3 
2.3 

We  thus  see  that,  if  the  A;th  power  of  a  +  6  follows  the  rule, 
then  the  (A:-|-l)st  power  also  follows  the  rule,  since  the  ex- 
l)ansion  of  (a  +  6)'=+^  as  just  found,  has  li -^\  everywhere  in 
place  of  A:  in  the  expansion  of  (a  4-6)*-;  that  is,  both  proceed 
according  to  the  same  law. 


THE   BINOMIAL   FORMULA  311 

Hence,  if  the  rule  for  the  binomial  formula  holds  for  any 
given  exponent,  it  also  holds  for  the  next  higher  exponent. 

But  we  have  found  by  actual  multiplication  that  the  rule 
holds  for  the  third  power.  Hence  we  know  by  the  above 
argument  that  it  holds  for  the  fourth  power,  even  without 
multiplying  it  out. 

But  the  argument  shows  that  if  the  rule  holds  for  the  fourth 
power,  then  it  does  for  the  fifth,  and  if  for  the  fifth  then  for 
the  sixth,  and  so  on,  as  far  as  we  please. 

Hence  we  say  that  the  Binomial  Formula  holds  for  any  posi- 
tive, integral  exponent  n.     Stated  as  a  formula  this  is: 

(a +  6)"  =  a"-h  na"-'b  +  "Al^  a"-'b^  +  n{n-l){n-2)  ^„_,^, 

n(n-l)(n-2)(n-3)       ,,,       ,^^ 
"^  234  ■ 

WRITTEN  EXERCISES 

1.  Write  out  the  fifth  term  in  the  expansion  of  (a  +  5)*. 

2.  Multiply 

2.3.4  ^  2.3 

by  h  and  add  the  products. 

222.  The  General  Term.  We  can  easily  make  a  rule  for 
writing  the  rtli  term  after  the  first.  For  instance,  in  the  4th 
term  after  the  first,  the  exponent  of  h  is  4,  the  exponent  of  a  is 
n  —  4,  the  last  factor  in  the  denominator  is  4,  and  the  number 
of  factors  in  the  numerator  is  4. 
Hence  the  rth  term  after  the  first 

^  n(n  -  l){n  -  2)  ♦.■  to  (r  factors)         ,, 
2'3-r 
E.g.  the  5th  term  after  the  first  in  the  expansion  of  (a  +  by  is 
?         2 
^•^•"^•^•^  a^-sfes  =  126  a^b^ 


312  THE   BINOMIAL   FORMULA 

Example  1.     Expand  (2x-\-  yf  by  the  binomial  formula. 

Solution.  (2x-\-  yf  =  (2  xy  +  3(2  xYy  +  3(2  x)y^  +  y« 

=  8  x3  +  12  x^y  +  6  y2  _|.  2/8. 

Example  2.     Expand  Ix^ j   by  the  binomial  formula. 

Solution,     (x^-  iy  =  (sc^y  -  l{x'^'f  •  -  +  ^lix^yl-Y  -  35(a;2)Viy 

=  a;"- 7  a;ii +  21  x8  -  35a;5  +  35  a;2  _  21  _^^ _  J_ 

a;      x^     X' 

WRITTEN  EXERCISES 

Expand  the  following  by  the  binomial  formula : 

1.  {x^yf.  5.    {2x-y)}  9.    {x-2yy. 

2.  {x-y)\  6.    (a;  +  22/)3.  lO.    {x-2yf. 

3.  (a;  +  ?/y.  7.    {x-2yf.  11.    (20^  +  32/)^ 

.  (.-,y.  8.  (a  +  lj.  la.  g4y. 

13.  Find  the  4th  term  after  the  first  in  the  expansion  of 
(x  4-  yf ;  of  (a;  -  y)^. 

14.  Find  the  5th  term  after  the  first  in  (2x-\-3  yy\ 

1\9 


15.    Find  the  6th  term  in  the  expansion  of  ix^ 


X 


16.  Find  the  5th  term  in  the  expansion  of  f  —  +^  )  • 

\y    ^J 

1  2 

17.  Find  the  7th  term  in  the  expansion  of  {x^  —  x'^y. 

18.  Find  the  6th  term  in  the  expansion  oi[x^ )  . 

19.  Find  the  7th  term  in  the  expansion  of  (  «^+-t j  • 


20.    Find  the  6th  term  in  the  expansion  of  (  a 

a? 


,,i-i 


10 


REVIEW   QUESTIONS  313 

HISTORICAL  NOTE 

The  Binomial  Formula.  Special  cases  of  the  binomial  formula  were 
known  very  early.  The  Hindus  and  the  Arabs  knew  the  formula  for 
(a  +  &)2  and  (a -\-by^  and  Vieta  (1591)  knew  it  for  (a  +  6)6.  Blaise 
Pascal  (see  next  page)  constructed  an  "arithmetical  triangle"  from 
which  the  coefficients  of  any  power  of  a  ~t-  6  could  be  read  when  the  next 
lower  was  given.  The  discovery  of  the  general  binomial  theorem  would 
seem  to  have  been  unavoidable  from  what  was  already  known.  It  never- 
theless remained  for  Sir  Isaac  Newton  in  1676  to  discover  this  remarkable 
theorem  and  to  make  use  of  it  in  many  important  applications. 

REVIEW  QUESTIONS 

1.  In  the  expansion  of  (a  +  by  by  the  binomial  formula  (1), 
what  are  the  exponents  of  a?  of  6  ?  What  are  the  successive 
coefficients  ? 

2.  What  is  the  rth  term  after  the  first  in  the  expansion  of 
(a  +  by  ? 

3.  How  does  the  expansion  of  (a  —  by  differ  from  that  of 
(a  +  by  ? 

4.  In  the  proof  of  the  binomial  formula  by  mathematical 
inductions, 

(a)  What  is  the  first  step  ? 

(6)  How  is  (a  +  6)*+^  derived  from  (a  -j-  bf  ? 

(c)  How  do  the  terms  of  (a  +  6)*'''^  compare  with  those  of 
(a  +  by  ? 

(d)  If  (a  +  by  follows  the  rule,  what  do  we  conclude  in 
regard  to  (a  -\-  by^^  ? 

(e)  Hence  if  the  rule  holds  for  (a  -f  by,  what  do  we  know  of 
(a-\-by? 

(f)  When  we  know  that  (a  -f  by  follows  the  rule,  what  do 
we  conclude  about  (a  +  6)^? 

(g)  How  does  this  argument  go  on  ? 


CHAPTER   XXI 
VARIABLES  AND  FUNCTIONS* 

223.  Definitions.  In  the  graph  of  the  equation  y  =  2xwe  may- 
think  of  the  point  P  represented  by  (x,  y)  as  moving  along 
the  line.  As  the  point  moves  the  values  of  x  and  y  both  change, 
subject  to  the  fixed  relation  that  y  is  always  twice  as  great  as  x. 

Variable.  An  algebraic  expression  whose  value  changes  in 
any  one  problem  or  discussion  is  called  a  variable. 

E.g.     As  P  moves  along  the  line  y=2  sc,  both  x  and  y  are  variables. 

Function.  When  two  variables  are  connected  by  a  fixed  re- 
lation so  that  the  value  of  one  depends  on  the  value  of  the 
other,  then  one  variable  is  said  to  be  a  function  of  the  other. 


' 

J 

I 

^j 

\^ 

K 

U} 

r 

J 

/ 

I 

1 

> 

f 

1 

^ 

f 

'y 

i' 

f 

' 

/ 

f 

/ 

L 

_ 

_ 

__ 

, 

_J 

s 

s 

s 

s 

s 

s 

F 

y 

( 

X' 

i 

) 

\ 

s 

s 

's 

S 

.fy 

\ 

'^ 

> 

^' 

h 

\ 

\ 

\ 

\ 

\ 

i_ 

^ 

E.g.    In  the  equation  y  =  2x,  if  any  value  is  given  to  x,  this  deter- 
mines the  corresponding  value  of  y.     Hence  y  is  a  function  of  x. 

Increasing  Function.     In  the  equation  y  =  2  x,  y  is  said  to  be 
an  iricreasing  function  of  x,  because  y  increases  as  x  increases. 


*  Articles  223-226  may  be  omitted  without  destroying  the  continuity. 

314 


Blaise  Pascal  was  born  at  Clermont,  France,  in  1623.  and  died 
at  Paris  in  1662.  He  was  a  celebrated  philosopher  and  mathema- 
tician. 

At  the  age  of  twelve  years,  Pascal  was  found  drawing  charcoal 
figures  on  the  pavement  and  proving  theorems  in  geometry.  At 
the  age  of  sixteen  he  wrots  a  treatise  on  algebraic  geometry  which 
displayed  great  power.  At  nineteen  he  invented  a  machine  for 
performing  arithmetical  operations,  —  the  forerunner  of  our  modern 
calculating  machines. 

In  1653  Pascal  devised  a  scheme  for  writing  down  the  coefficients 
of  the  binominal  expansion,  which  he  called  the  "  Arithmetical 
Triangle,"  and  which  has  ever  since  been  known  as  the  "  Pascal 
Triangle." 

Pascal  wrote  extensively  on  both  philosophy  and  mathematics. 


VARIABLES  AND   FUNCTIONS  315 

Decreasing  Function.  In  the  equation  y  =  1  —  x,  y  is  said  to 
be  a  decreasing  function  of  x,  because  y  decreases  as  x  increases. 

Eg.     If  X  increases  from  0  to  1,  ?/  decreases  from  1  to  0. 

The  graphs  of  ?/  =  2  x  and  y  =  1  —  x  show  the  relations  of 
the  variables. 

E.g.  Think  of  the  values  of  x  as  increasing  from  0  toward  the  right. 
Then  my  =  2x  the  values  of  y  ascend.^  while  my  =\  —  x  the  values  of  y 
descend. 

ORAL  EXERCISES 

1.  A  man  walks  at  a  uniform  rate  of  3  miles  an  hour. 
Using  d  for  the  distance  walked  and  t  for  the  number  of  hours, 
express  d  as  a  function  of  t. 

Is  d  an  increasing  or  decreasing  function  of  ^  ? 

2.  A  train  moves  at  a  uniform  speed  of  30  miles  an  hour. 
Express  the  distance  it  moves  as  a  function  of  the  time. 

3.  A  pipe  pours  out  24  gallons  of  water  a  minute.  Express 
the  amount  poured  out  as  a  function  of  the  time. 

224.  Definition.  If  A;  is  a  constant,  and  \i  y  =  kx,  then  y  is 
said  to  vary  as  x.  This  is  written  yocx,  and  is  read  y  varies 
as  X. 

WRITTEN   EXERCISES  AND  PROBLEMS 

1.  If  y  OCX  and  y  =  S  when  x  =  o,  find  y  when  x  =  9. 

Solution.     Let  y  =  kx.    If  ?/  =  3  when  a:  =  5,  then  3  =  ^.5  and  k  =  ^. 
Hence,  when  x  =  9,  y  =  kx  =  ^  •  9  =  ^-  =  5^. 

2.  If  y  oc  X  and  y  =  7  when  x  =  3,  find  y  when  x  =  7 

3.  If  ?/  oc  X  and  y  =  a  when  x  =  b,  find  y  when  x  =  c. 

4.  The  circumference  of  a  circle  varies  as  the  diameter,  that 
is,  c  =  IT  '  d.  Eind  the  circumference  of  a  circle  whose  diameter 
is  10  feet  if  a  circle  whose  diameter  is  1  foot  has  a  circumfer- 
ence of  approximately  3.1416  feet. 


316  VARIABLES  AND  FUNCTIONS 

5.  The  interest  on  a  fixed  sum  of  moDey  varies  as  the  length 
of  time  it  is  invested.  If  a  certain  sum  draws  $  145  in  three 
months,  how  much  will  it  draw  in  11  months  ? 

6.  The  interest  on  a  sum  of  money  invested  for  a  fixed  time 
at  a  fixed  rate  varies  as  the  amount  invested.  If  the  interest 
on  $  800  is  $  125,  what  will  be  the  interest  on  $  540  at  the 
same  rate  and  for  the  same  time  ? 

7.  If  a  steamer  uses  340  tons  of  coal  in  going  425  miles, 
how  much  will  it  use  in  going  3240  miles  ? 

8.  If  an  automobile  uses  8i  gallons  of  gasoline  in  going  96 
miles,  how  many  gallons  will  it  use  in  going  370  miles  ?  How 
far  will  it  go  in  using  32  gallons? 

9.  The  weight  of  an  object  below  the  earth's  surface  varies 
directly  as  the  distance  from  the  earth's  center.  If  an  object 
weighs  100  pounds  at  the  earth's  surface  (4000  miles  from  the 
center),  what  will  it  weigh  2000  miles  from  the  center  ?  500 
miles  from  the  center  ?  How  far  from  the  center  must  it  be 
to  weigh  just  one  pound  ? 

225.  Inverse  Variation.  If  the  number  of  men  emplo3'ed  on 
a  certain  piece  of  work  is  doubled,  the  time  required  to  finish 
the  work  is  divided  by  2.  If  the  number  of  men  is  trebled, 
the  time  is  divided  by  3,  and  so  on. 

If  n  is  the  number  of  men  and  t  the  number  of  days  required 
to  finish  a  piece  of  work,  then  the  relation  described  above  may 

be  expressed  by  t  =  - ,  where  k  is  some  fixed  number  depend- 

n 

ing  on  the  piece  of  work  to  be  done. 

k  .     • 

When  /  =  - ,  we  say  that  t  varies  inversely  as  n,  and  this  is 

also  written  ^  oc  -  • 
n 


VARIABLES  AND  FUNCTIONS  317 

EXERCISES 

Give  the  results  orally  in  Examples  1  to  4. 

1.  If  « =  -,  and  if  ?i  =  4  when  t  =  2,  find  k, 

n 

2.  If  ^  =  -,  and  if  ?i  =  7  when  t  =  3,  find  k. 

n 

.3.    If  ^  =  -,  and  if  n  =  15  when  t  =  \,  find  k. 
n 

4.  If  ^  =  -,  and  if  ?i  =  i  when  t  =  21,  find  fc. 

n 

5.  If  a  piece  of  work  can  be  finished  in  84  days  by  16  men, 
how  long  will  it  take  40  men  to  do  it  ? 

6.  If  126  men  can  do  a  piece  of  work  in  80  days,  how  many 
men  will  be  required  to  do  this  work  in  120  days  ? 

226.  Variation  as  the  Square  of  a  Variable.  We  have  seen 
(page  261)  that  a  =  irv'^  where  r  is  the  radius  of  a  circle  and  a 
its  area.     We  say  that  a  varies  directly  as  r^. 

It  follows  at  once  that  the  weight  of  a  circular  disc  cut  from 
a  sheet  of  metal  of  even  thickness  varies  as  the  square  of  the 
diameter. 

WRITTEN  EXERCISES 

1.  If  IV  =  kr"^,  and  if  w  =  10  when  r  =  1,  find  iv  when  r  =  1^. 

2.  If  2/  =  T^'^'^i  and  if  ?/  =  4  when  x  =  2,  find  k.  Also  find 
y  when  x  =  4. 

3.  If  y  =  kx"^^  and  if  ?/  =  1  when  x  =  3,  find  ?/  when  x  =  15. 

4.  A  circular  piece  of  metal  one  foot  in  diameter  weighs 
8  pounds.  How  much  will  a  circular  piece  weigh  if  it  is  cut 
from  the  same  sheet  of  metal  and  is  2^  feet  in  diameter? 

5.  The  distance  passed  over  by  a  falling  body  varies 
directly  as  the  square  of  the  time. 

If  a  falling  body  goes  16  feet  in  one  second,  how  far  will 
it  go  in  7  seconds  ? 


CHAPTER   XXII 
REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  I  * 

1.  Write  a  formula  for  the  area  of  a  rectangle. 

2.  Write  a  formula  for  the  volume  of  a  rectangular  solid. 

3.  If  a  =  2,  b  =  3,  c  =  4,  find  the  value  of 

7    ,  o7>         ,  4^  +  c     3c-2b 

6  +  c  —  a;  J:o  —  c-\-a;   — ; 

CI  a 

4.  Perform  the  indicated  operations  : 

20k-{-10k-16k',  15  71  -I-  8  71  -  3  71 ; 

50 ax  —  20  ax  —  7 ax;  63  ac  +  7  ac  —  50 ac. 

5.  Multiply  a  +  5  -f-  c  by  X ;  m  —  ti  —  2  by  a  ;  x  —  y  —  zhy  4:. 

6.  Divide  42  +  12  by  3  without  first  adding. 

7.  Divide  12  a;  —  8  ?/  +  16  2  by  4  ;  6  ra  +  12  sx  by  6  x. 

8.  Multiply  3  •  4  •  5  by  2  in  four  different  ways. 

9.  Multiply  6  ax  by  4  ;  10  abc  hy  d;  16  xyz  by  3  ;  8  ??i7i  by  7. 

10.  Divide  4  •  8  •  12  by  2  in  three  different  ways. 

11.  Divide  10 abc  by  5  c;   18  m?^  by  9;   12  xy  hj  S.     Divide 
10(2  a;  4-  4  2/  +  6  2!)  by  2  in  two  different  ways. 

12.  Find  the  value  of  3-2H-4-5  —  2-7.     What  operations 
are  performed  first  ? 

1 3.  Find  the  value  of  3(2  +  8  -  4)  -  2(6  -  1 ). 

14.  If  «  =  2,  b  =  1,  c  =  3,  d  =  4,  find  the  value  of  each  of 
the  following : 

c.d^a-2b;  Gabc-^2d',  4.  ad-3(c-2  b) ;  {10ac-2b)^d. 

318 


KEVIEW  EXERCISES  319 

REVIEW  EXERCISES   FOR   CHAPTER  II 

1.  Solve  orally  the  equations  : 

2  a;  +  3  a;  =  20 ;  2(3  a;  -  a;)  =  8 ; 

16  2/  -  2(3  ?/  +  2/)  =  24 ;  in-^3n-n=  15. 

Solve  the  following  equations  :  • 

2.  4  ?i  4  5  ?i  +  6  =  3  n  -j- 18. 

3.  2  a;  +  4  :r  —  2  a.'  =  6  a;  —  3  a;  +  15. 

4.  7(a;  +  2)+  2(4  x  _  1)  +  G  =  6(x  +  4)4-3(2  x  -  1). 

5.  Six  times  a  certain  number  plus  12  equals  42.     What  is 
the  number  ? 

6.  For  how  many  years  must  S  5000  be  invested  at  G  %  to 
yield  $1800  interest? 

7.  The  greater  of   two  numbers  is  8  times  the  less,  and 
their  sum  is  90.     Find  the  numbers. 

8.  Find  a  number  such  that  when  5  times  the  number  is 
subtracted  from  12  times  the  number  the  remainder  is  42. 

9.  Find  three  consecutive  integers  such  that  4  times  the 
first  is  7  greater  than  3  times  the  third. 

10.  A  rectangle  is  twice  as  long  as  it  is  wide.  The  perimeter 
exceeds  the  length  by  40.     Find  the  dimensions. 

11.  The  area  of  Louisiana  is  (nearly)  4  times  that  of  Mary- 
land, and  the  sum  of  their  areas  is  60,930  square  miles.  Find 
the  (approximate)  area  of  each  state. 

12.  The  horse  power  of  a  certain  steam  yacht  is  12  times 
that  of  a  motor  boat.  The  sum  of  their  horse  powers  is  195. 
Find  the  horse  power  of  each. 

13.  At  a  football  game  there  were  2000  persons.  The  num- 
ber of  women  was  3  times  the  number  of  children,  and  the 
number  of  men  was  6  times  the  number  of  children.  How 
many  men,  women,  and  children  were  there  ? 


320  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  III 

Perform  the  following  indicated  operations : 

1.  _8-(-9)-(-h7)  +  8. 

2.  l6+(-18)-(+2)4-4. 

3.  i2x-\-(-7x)-{-3x)-\-2x. 

4.  Un-\-{-S0n)-{-7  7i). 

5.  8.(-3).(-l).  8.    (_12)H-[(-2)-(-2)]. 

6.  [48-(-6)]^(-l).  9.    (-42).(-2)-(-7). 

7.  3.(-2)(-3)(-l).  10.    [(7a6)-(-a)].(-3). 

11.  What  is  meant  by  the  average  of  several  numbers  ? 

12.  Find  the  average  of  20,  16,  8,  4,  0,  -  8,  -  12. 

13.  Find  the  average  of  -  8,  -  32,  14,  26,  -  40. 

Solve  the  following : 

14.  .T  +  6  =  4.  17.  3a;-8  =  -16. 

15.  3aj  +  12  =  6.  18.  2{x-{-6)  =  3(x-^5). 

16.  S  +  x  =  4:.  19.  2a;  +  4  =  3a;  +  8. 

20.  5.T4-7-2ic  =  2a5  +  9. 

21.  3?i  +  2(?i  +  4)  =  4n  +  14. 

22.  8  +  6a;  +  3(2  +  a;)=5a;+26. 

23.  4(x'  +  3)+  2(3 a;  +  1)=  5(x-\-  2)  +  2(ic  +  3)+  19. 

24.  7(a;  +  l)  +  3(2a;  +  3)  =  4(iC+5)  +  7(a;  +  2)-8. 

25.  7(2a;-3)  +  5(4a;-l)=3(a.-+l)  +  2. 

26.  If  —  n  represents  a  negative  integer,  how  do  you  repre- 
sent the  next  integer  to  the  right  on  the  number  scale  ?  the 
next  to  the  left  ? 

27.  If  —2n  represents  a  negative  even  integer,  how  do  you 
represent  the  next  even  integer  to  the  right  ?  the  next  to  the 
left? 

28.  Do  negative  numbers  apply  to  all  things  to  which  posi- 
tive numbers  apply  ?  Can  there  be  a  negative  number  of  books 
on  a  shelf  ? 


REVIEW   EXERCISES  321 

REVIEW  EXERCISES  FOR  CHAPTER  IV 

1.  Add  Sx-\-4:y  —  3z,  5  x  —  2  y  —  z,  and  3y  —  5x-\-7z. 

2.  From  15  a  +  4  6  —  13  6c  subtract  3  a  —  86  -f  2  6c. 

3.  Subtract  7  x  —  5y  —  la  from  6  a;  +  5  y  +  3  a. 

4.  From  5ic  —  4?/  —  92:  subtract  2t  x  —  ^  y  -\-  2  z. 

5.  Add  5  a  +  3  6  —  2  c  and  11  a  —  7  6  -f  8  c. 

6.  Add  11  axy  -f  13  a;  —  14  ?/,  2  ?/  —  4  x,  and  3  ?/  +  x  —  8  axy. 

7.  Simplify  (5  a^  -  3  6)  +  (2  a;  +  6)  -  (4  a.-  -  2  6  -  x  +  5  6). 

8.  Add  19  6  +  3  c,  2  6  -  7  c,  2  c  -  14  6,  and  c  +  8  6. 

9.  Simplify  —  (a  —  3  6  —  c)  -  (2  c  —  a  —  5  6)  +  (a  —  c  +  6). 

10.  Subtract  2  a;  +  4  2/  +  2  from  13  .r  —  3  ?/  —  5  2;  +  8. 

11.  Solve  5(a;  -  7)  +  3(14  -  x)  +  60  =  1  -  10  a;. 

12.  Solve  13(1 -.!■)  -6  (2  a;- 5)  =  80 +  12  a;. 

13.  Add  7a;  —  3^/  —  4,  5  a; +  22/4- 5,  and  3  y  —  S  x  —  6. 

14.  Add  13  a  +  4  6  -  9  c,  2  c  -  8  6  -  16  a,  and  8  a  -  5  6  -  8  c. 

15.  Simplify  8  a;  -  [2  a;  +  3  (a;  -  1)  -  (2  x  -  3)]. 

16.  From  17  6  —  4  a  —  2  c  —  19  subtract  8  c  —  5  a  —8  6  +  4. 

17.  Simplify  3  -  (3  -  2  +  6  +  8  -  3)  +  8  -  (9  -  3  +  8). 

18.  Solve  3(4  _  a:)  -  2  (5  -  6  a-)  =  8  a;  +  4. 

19.  Simplify  12  +  (2  a  -  3  c  -  4  6)  -  (3  6  -  c  -  a  -  8). 

20.  Simplify  5  a;  -  (3  x  -  2  +  2  y  +  a^)  +  13  2/  -  (6  -  3  a;). 

21.  Add  2/  -  20,  4  2/  H-  6,  2  2/  +  4  .X-  -  13,  and  2  a;  -  8  ?/  -  40. 

22.  Subtract  16  —  x-\-2z  —  -iy  from  3x  —  oz  —  Sy. 

23.  Solve  19  +  (2  a;  -  7)  -  (31  -  4  a;  -  8  -  2  a-)  =  5  a;  -h  7. 

24.  Solve  16-1-5  x  -  (8  x  -f  9  -  4  x.-\- 17)  =Sx-  3. 

25.  Solve  6  a:  -  3  -  (4  .X-  +  8  -  9  x)  -  (5  .i-  -  2)  =  x  +  11. 

26.  The  sum  of  two  numbers  is  16.  Seven  times  one  is  S- 
less  than  5  times  tlie  other.     What  are  the  numbers  ? 

27.  From  a  certain  number  a  there  is  subtracted  3  times  the 
remainder  when  8  is  subtracted  from  2  a.  Express  the  result 
in  terms  of  a. 


322  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  V 

Multiply  as  indicated : 

1.  (x-^l)(x-2).  3.    (2x-y-^l)(x-y), 

2.  (a-}-b)(2a  +  b).  4.    (S  x-3y)(2x -\- 5y). 

5.  (a  — 1  +  ^  — c-d)(4a  +  5&  +  3c-2d). 

6.  (4  ax  —  3  ay  -\-  5  az  —  S){x  -^  y  —  z  +  2). 

7.  (3  a  -  2  6  +  4  c)(2  a  +  3  6  -  c). 

Divide  as  indicated : 

8.  a'  -  12  «2  4-  27  a  +  40  by  a  -  5. 

9.  a;5 — 5  .x-''?/ +11  x^y"^ — 14  a.-^^/^  -|-  9  xi/^  _  2  ?/5  by  x^ — 3  a;?/ + 2  2/^^. 

10.  .1'*  +  xhf  +  2/"  by  x^  —  xy  +  2/1 

11.  a^  +  5  rt2  _  2  a  -  24  by  a^  +  7  a  +  12. 

12.  a'  -5a'b  +  10 aW  -  10  a'^b^  +  5  afe*-  b'  by  a2-2  a5+  61 

13.  x^  —  5  a^2/^  —  5  a;2?/3  -(-  ?/5  by  x"^  —  3  .t?/  +  ?/^. 

14.  Add  12  a252c  _j_  g  <^^.^  6  aa;  —  8  a262c,  and  2  aa;  +  3  a'^b^c. 

15.  Add  5  a'?/2  +  3  a."^^/  +  ^  a.'?/,   2  a;^?/  -~  ^  ^'Z/^  —  ^  ^I/)  ^.nd  4  a;?/. 

16.  Add  6  ab  —  S  c  —  2  a,  2  c  —  4:ab  —5  a,  5  c  —  a  -\-  ab. 

1 7 .  From  35  «5 — 8  a;  —  9  ;2  +  13  subtract  16  ab  —  4:Z -{- 5  x-\-S. 

18.  Subtract  oa  —  Sx  —  6y  from  13 a;  +  14 ?/  —  15 2;  —  4 a. 

19.  From  9y  —  4.x  —  6z-3b  subtract  8  -  9  ?/  —  3  x  —  2z. 

20.  Solve  (n  -  4)(6  -  3  n)  -  (6  -  7iy  -  10  =  -  4  n(n  -  4). 

21.  (71  +  2)2  -f  (n  -  1)2  +  (n  4- 1)2=  3  n{n  +  2)  +  60  n  +  130. 

22.  2  a-  +  4  -  6(5  a;  -  8  -  7  a;)  +  2  -  4  a.-  =  6(2-3  a^  -  42. 

23.  A  man  bought  a  tract  of  coal  land  and  sold  it  a  month 
later  for  S  93,840.  If  his  gain  was  at  the  rate  of  24  %  per 
annum,  what  did  he  pay  for  the  land  ? 

24.  The  melting  point  of  copper  is  250  degrees  (Centigrade) 
lower  than  4  times  that  of  lead.  Ten  times  the  number  of 
degrees  at  whicli  lead  melts  minus  twice  the  number  at  which 
copper  melts  equals  1152.     Find  the  melting  point  of  each. 


REVIEW   EXERCISES  323 

REVIEW   EXERCISES  FOR  CHAPTER  VI 

Use  the  formulas  for  (a  ±  by  for  the  following : 

1.  [(a  +  6)  +  (c  -  rf)]2.  4.    [7  a;  -  (4  r  -  .s)]2. 

2.  [(a  ^  3)  -  (^  +  c)  J.  5.    [(m2-3)-LVm'^  +  ?0]'' 

3.  [(3a-2&)+5]2.  6.    [3(2  +  y)  -  2(3  +  a;)]2. 

Use  the  formula  for  (a  -f  6)(a  —  Z>),  for  the  following; 

7.  [a  4-  6  +  (c  -  d)']la  +  h-{c-  cZ)]. 

8.  [a;  +  2/  +  (w  +  v)]  C-^*  +  i/  -  (''  +  ^V]- 

9.  [4  a;  -  (a  -  2  6)]  [4a;  +  {a  -  2  6)]. 

10.  [a  +  2  6  -  (a;  -  if)J_a  +  2  6  +  (.r  -  2/^)]. 

11.  {lllfx-Zhx'')(llh^x  +  ^h:^). 

Find  the  following  as  indicated : 

12.  (2  a;  —  3  ?/  -f  ;^)2.  14.    (o?  +  f)  ^  {x  +  y), 

13.  (2.^•-3?/)^  15.    (8rt3_l25  63)--(2a-5  6). 

16.  16  c  -  (41  -  7  c)  +  (15  -  8  c). 

17.  -  (5  a  -  3  c)  -  (2  c  -  8  a)  +  3  a. 

18.  -  (-  12  a;  -  7  2/  -  15  a-)  -  (-  9  2/  H-  8  .X-  +  3  y). 

19.  (19  a;  +  4  ?/  -  32  a;  -  17  a;)  -  12  a;  -  (49  ?/  +  18  x'  -  70  x). 

Solve  the  following  equations  : 

20.  7(m  +  6)  +  10  m  =  42  -  8(2  m  +  2)  +  181. 

21.  20 -3(a;- 4) +2  a;  =  2  a; +  17. 

22-   6(a;  -  3)  -  2(2  -  a;)  =  2(.«  +  l)- 6. 

23.  If  two  numbers  differ  by  d  and  if  the  greater  of  the 
numbers  is  x,  how  do  you  represent  the  other  ? 

24.  A  father  is  3  times  as  old  now  as  his  son  ^vas  7  years 
ago.  If  the  son's  age  now  is  represented  by  a;,  how  is  the 
father's  age  represented  ? 

25.  A  picture  inside  the  frame  is  w  inches  wide  and  w  4-  6 
inches  long.  The  frame  is  4  inches  wide.  Express  the  area 
of  the  frame  in  terms  of  lo. 


324  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  VU 

Factor : 

1.  0.-2  + 5x  +  4.  9.  W-27.  17.  :x^-\-f. 

2.  x^-ox  +  4..  10.  ^o?-h\  18.  125a3+5^ 
Z.  a'-la^-\-12.  11.  l  +  64af'.  19.  1  +  125  a^^. 

4.  a«  -  a'' -  12.  12.  a^  -  6^  20.  27  ic^  -  1. 

5.  6-4-3  5-18.  13.  l-64a^.  21.  1  -  8  x^^'. 

6.  6^-Z>2-56.  14.  ?(;«  +  27a«.  22.  1  +  Sa?f. 

7.  a3+8.  15.  ^(;6-8a^  23.  8  ar' +  27 /. 

8.  27a3  4-6l  16.  27  a^  -  8  ?>^  24.  8  a.-^  -  27  .v". 

25.  c'' -  31  c2  +  220.  27.    26  +  399i-22m-33mw. 

26.  ac  ^- d'^a  -  Wc  -  ¥d\         28.    12  a;^  +  11  a;  -  56. 

29.  a2  +  4  a6  +  4  62_  (a2  _  4  a6  +  4  W). 

30.  (3  a?  -  1)2- (a:2  +  4  ^2  _  4  r^y^ 

31.  (a:  +  3  yy  +  {x  -2yyJr2(x  +  Sy){x-2  y). 

32.  16 (a  +  6)2-8  (a  -  6)(a  -\-h)-\-(a-  h)\ 

33.  256  aj2  _  (49  x''-\-  4  ?/»-  28  a.^). 

Factor  the  following : 

34.  a"^  4-  4  a6  +  4  h"^—  {3?—  2  xy  -{-  7/). 

35.  (3  X  -  2)2-  (4  x'  +  9  ^2  _  12  xy). 

36.  .i"2  4-4a;^  +  4  2/2— (a2  4-2a6+ 62). 

37.  16  a;y- (4  a;2  +  9?/2+  12  xy.) 

38.  (a  +  5)'-(4a2  +  9  62-12a6). 

39.  a4  +  a262  +  6^  40.    16  a;^  +  20  a;22/2  +  9  y*. 

41.  The  Nile  is  100  miles  more  than  twice  as  long  as  the 
Danube.  Ten  times  the  length  of  the  Danube  minus  4  times  the 
length  of  the  Nile  equals  3400  miles.     How  long  is  each  river  ? 

42.  Lead  weiglis  259  pounds  more  per  cubic  foot  than  cast 
iron,  and  166  })ounds  more  than  bronze;  while  a  cubic  foot  of 
bronze  weighs  807  pounds  less  than  .3  cubic  feet  of  iron.  Find 
the  weight  per  cubic  foot  of  each  metal. 


REVIEW    EXERCISES  325 

REVIEW  EXERCISES  FOR  CHAPTER  VIU 

Solve  the  equations : 

1.  (x-l){x  +  l){x-S)=0, 

2.  x(x''-4:)(x''-Q)  =  0. 

3.  0^2  +  7  X  +  12  =  0. 

4.  a^  +  3a'2-4ci'-12  =  0. 

5.  a^-Sx''-4.x  +  12  =  0. 

6.  (a;  -  1)(2  a;  -  2)  +  (^  -  5)2  ={S-x) (24  -Sx)-T. 

7.  (17a:  +  3)(a;-l)+8  =  (2-.T)(6-17a;)  +  19. 

Perform  the  operations  indicated : 

8.  (a  -  2)(6  a  -  4)  +  2(a  -  1)^  =  (6  -  a)(30  -  8  a)  +  4. 

9.  5  _  (a  +  ?>  -  c  -  d  +  8)  +  (3.+  a  +  c  -  cZ)  -  5. 

10.  Add  G  a  +  9,  8  a  -  13,  46  a  -  8,  and  6  -  54  a. 

11.  From  3  —  4a— 5c  +  8a;2  subtract  2x2  —  2a  —  4c  +  8. 

12.  (4a6  —  6ac  — 5afZ)(6  — c  +  c?). 

13.  From  6(a  + 2)  + 3(c  + 4)- 2(5- fZ) 

subtract  2(a  +  2)  -  2(c  +  4)  +  3(6  -  d). 

14.  The  sum  of  two  numbers  is  13  and  the  sum  of  their 
squares  is  97.     Find  the  numbers. 

15.  The  sum  of  two  numbers  is  38  and  their  product  is  240. 
Find  the  numbers. 

16.  The  sum  of  two  numbers  is  23  and  their  product  is  120. 
Find  the  numbers. 

17.  The  difference  between  two  numbers  is  14  and  their 
product  is  176.     Find  the  numbers. 

18.  A  rectangular  field  is  10  rods  longer  than  it  is  wide,  and 
its  area  is  1200  square  rods.     Find  its  dimensions. 

19.  An  open  box  is  made  from  a  square  piece  of  tin  by  cut- 
ting out  a  6-inch  square  from  each  corner  and  turning  up  the 
sides.  How  large  is  the  original  square  if  the  box  contains 
150  cubic  inches  ? 


326  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  IX 

Find  the  H.  C.  F.  of  the  following : 

1.  X'  —  y"^,   a^  —  y^,   x^  —  2xy-{-y^. 

2.  x^  +  2  xy  +  y"^,    x^  -f  y^,   x^  +  xy. 

3.  a2 H-  7  a  +  12,   a" -  4,   a"-  9. 

4.  125 +  7«^   25 -m2,    m2  +  10m  +  25. 

Find  the  L.  C.  M.  of  the  following: 

5.  .^2  -  11  a.' +  30,   a;2-36,   x''-2d. 

6.  a''- a- 6,   a2-7aH-12,    a2-2a-8. 

7.  a-3,    ci2_|_3c(+9,   a3-27. 

8.  x'^-3x  +  2,   ic2-5a;  +  6,   a;2-4a;  +  3. 

Factor  the  following : 

9.  (^2-xy-2(;2-x)(x-i)-\- {x-iy. 

10.  {2 -\-yy +  2(2 +  y)(l  +  x)+(l-\-xy. 

11.  (3a  -  2  &)2-10(3a- 2  6) +  25. 

12.  (6  a  -  by  +  (2  a  +  1)^  -  2(6  a-b){2a  +  1). 

13.  25(a  +  by  +  50(a  +  b)(a  -b)  +  25(a  -  by. 

14.  aj2_|.i2fl;(a  +  &  +  c)4-36(a  +  ^H-c)2. 

15.  49(m  -  Sy  +  36(m  +  1)^  -  84(m  -  3)2(m  + 1)1 

16.  16(.«  -  yy  -  16{x  -  2/)(a;  +  2/)  +  4(.i^  +  2/)'- 

17.  -  30(a  +  b){a  -  by  +  25(rt  -  6)^  +  9(a  +  by. 

18.  Express  the  average  of  the  numbers  3,  8,  —9,  12;  also 
of  the  numbers  3  a,  6,  2  c,  —5  b. 

19.  Express  the  sum  of  the  squares  of  four  consecutive  even 
integers  of  which  2  7i  is  the  smallest. 

20.  Express  the  sum  of  the  squares  of  four  consecutive  odd 
integers  of  which  2  ?i-+-l  is  the  greatest. 

21.  A  wall  I  feet  long  and  h  feet  high  has  three  windows 
each  k  feet  wide  and  ??i  feet  high.  By  how  much  does  the 
area  of  the  wall  exceed  that  of  the  windows  ? 


REVIEW    EXERCISES  327 


REVIEW  EXERCISES  FOR  CHAPTER  X 

Perform  the  following  indicated  operations : 


10. 


x-2      4-.t2      2  +  x 


(a  —  b)  {b  —  c)      (c  —  6)(c  —  a)      (a  —  c)(b  —  a) 

\  a-bj     \2a^-\-3ab  +  by 

a  -\-b  _a  —  b  ^  —  y      ^  +  2/ 

a  —  b      a  -\-b  x  y 

5.  "^ 


a^  +  b-  _  g-  —  6^  x  —  y      x  -\-  y 

a^  —  b'^      a-  +  6^  y  x 

[2         1       ,       1      ]        [a  +  aj     a  —  X 

{ , 1 •  \  -^ ; — 

[X     a  -\-  X     a  —  X  ]        [a  —  X     a  -\-  x 

a      J\  b      J     \b^  a 


X  —  y        x-\-yy  —  4:Z 


x-2z      x-{-2z      4:z''-x^ 


/        a^~^2ab  +  by         4ab  ^\ 

\  ab  J\a?-2ab  +  b''       J 


a? +  21  .  a'b  -3ab  +  ^b 
a^-S    '      a2  +  2  a  +  4  • 


11.     iy+.^y\{y-^lU\(lt-^' 


fi  _I_  .>.2 


y  —  xj\        x-\-yJ\y'  +  x 

12     A'-^      ^^'4-3\./     1  7 

^a'-3     a;H-4y  '  VaJ-fl     a;-3 

13.  The  world's  gold  production  in  1908  was  29  million 
dollars  less  than  3  times  that  of  1893,  and  the  production  in 
1900  was  59  million  less  than  twice  that  of  1893.  The  pro- 
duction of  1900  and  1908  together  amounted  to  697  million. 
How  much  was  produced  each  year  ? 


328  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  XI 

Solve : 

4     8      16     2      32 
4  10  4  * 

3  3  3  -^ 

.V^^i20     v  +  5^25. 
2  4  5 


5. 
6. 

7. 
8. 


y     2/  +  20     y-5      .V-10^-^5 
3  5  5  2 


iC  X 


x-1      x-\-l      (a;  — l)(a;  +  l) 
X  -1  ,  X  -\-l  27 


x  +  2     x-3      (a;  +  2)(a;-3) 
2a;-l      4a;-l  -10 


x-\-2       2x-'S      (x-{-2){2x-S) 


_     a  ,  a-^7     a  — 3     a  4- 227      ^ 
^-    S-^~i  3~^"~5  ^• 

10.  ^  +  ^  +  ^  =  2  +  a. 

11.  ^i_+l4.fiJIL?  +  ^Llll=2a-26. 

4  4  4 

,  „     71  -[-1  ,  71  -\-  3  ,  n  —  1      71  4- 13  ,  ??-  —  2 

1/5. = • 

344  33 

Perform  the  following  indicated  operations : 

13.  16a.T  +  4  -(8-  Sax-a)-(12ax-  13  -  ao;). 

14.  a25  -(36  -  8a2  -  7)-\-3ab'^  -(4: ah'  +  8  -  2a2). 

15.  Add  15  ax-"  +  3  be",  2  be'' -7  ax\  and  5  +  2  ax''  -  5  bc\ 

16.  Add  16-7«6-2a2  4.5a&,  4a2-2a6,  and  5a6-8. 

17.  Add  51  xhj  -35  +  12  a",  41  -  17  a^  -  57  xhj,  and  3  xh/. 


REVIEW   EXERCISES  329 

18.  How  do  you  represent  a  fraction  whose  numerator  is 

3  less  than  twice  that  of  —  and  whose  denominator  is  equal  to 

n 

3  times  the  sum  obtained  by  adding  2  to  the  denominator  of 

this  fraction  ? 

19.  There  are  two  numbers  such  that  if  one  half  their  prod- 
uct is  divided  by  twice  their  sum  the  result  is  12  times  their 
difference.  Write  an  equation  representing  this  relation  be- 
tween the  numbers. 


REVIEW  EXERCISES  FOR  CHAPTER  XII 

1.  If  ?  =  :^,  show  that  ^^±*=i±^. 

b      d  a  c 

2.  Which  is  the  greater  ratio,   —^ — ^  or  —^ — ^,  x  and  y 
both  being  positive  ?  -r    y       ^-r    y 

3.  Find  a  fourth  proportional  to  25,  75,  and  100. 

Solve : 

4.  (9x-  3)(4  -  x)-{-(x  -  3)2  =  _8(.r-f  2)2  +  94. 

5.  {x  -f  1)2  +  (aj  +  2)2+(a;  +  3)2  =(3 x  -  l)(x  + 12)-  43. 

6.  (2  X  +  5)  (x  -  7)  -  {x  -  1)2  =  {x  H-  1) {x  -f  2)  -  28. 

7.  3(5  -  xf  -{2x  -l){x  -l)  =  {x  -  l){x  +  10)+  17  a;  +  50. 

8.  (32  +  x){l  a;  -  l)  +  (5  -  xf-\-(x  -  If  =  6(x-\-iy-\- 194. 

9.  (2  X  -  7)(5  -  a;)  -  (2  -  5  x)  (1  -  x)  =  -  x{7  a;  -  34)  -  17. 

,^     x-\-S     x-9  ,  ;i--17      4a.--7  ,  2x  +  6  ,  5  -  31  x 
J.U. 1-  •  =  -f- \-  . . 

2  12  6  2312 

^,     3a;-l      3a;4-3  ,  x-1      x  +  5  ,    .         20 

11. =  ■ \-4:X -• 

6  3  2  6  3 

a;  —  ?>a;-ha_o 
1/5. 1 :. 

a  0 

,^     3.T-16  ,      21        6x-ll 
2       ^a;-8  4 


330  REVIEW   EXERCISES 

14.  How  may  the  signs  of  the  factors  of  a  product  be 
changed  without  changing  the  value  of  the  product?  Make 
all  possible  changes  of  sign  in  (a  —  b)(b  —  c)(c  —  d)  which  will 
not  change  its  value.  Also  make  all  possible  changes  of  sign 
which  will  change  the  sign  of  the  product. 

15.  State  how  the  signs  involved  in  a  fraction  may  be 
changed  without  changing  the  value  of  the  fraction.  Make 
all  possible  changes  of  signs  which  will  not  change  the  value 

or  -;  also  01 

16.  Make  three  changes  of  signs  each  leaving  unchanged  the 

value  of  the  expression  ^^ • 

^  (a  -  b)(b  -  c) 

17.  Reduce       ~    ,   ,  and    — to   fractions 

having  a  common  denominator. 

18.  Reduce ,  ,  and 


(a  -b)(b-cy  (c  -  d){b  -  a) '  (c  -  b){d  -  c) 

to  fractions  having  a  common  denominator. 

REVIEW  EXERCISES   FOR  CHAPTER  XIII 

1.  What  values  of  the  letters  in  an  identity  satisfy  it  ? 

2.  State  in  the  form  of  identities  as  many  as  possible  of 
the  eighteen  principles  given  in  this  book. 

In  Examples  3-12,  determine  which  are  identities : 

3.  {x-\-yy  =  x''-\-2xy-\-y\     5.    {x  +  iy  =  x''-2. 

4.  (x  —  yy  =  x'^  —  2xy-\-y\     6.    c^  +  ab -^  b- =  {a-\-by- ab. 

7.  a2-62=(a-6)(a  +  6). 

8.  (a  -  6)(a2  +  ab  +  b'-)  =  «■"'  -  b\ 

9.  («  +  b){a'^  -  a6  +  b^)  =  a^'  +  b\ 

10.  (x  +  b){x  +  (i)  =  x'^  +  (rt  +  h)x  +  ab. 

11.  (x  —  a){x  +  b)  =  x;^  —  (a  +  b)x  -\-  ab. 


REVIEW  EXERCISES  331 

12.  (a  +  6  -  c)2  =  a2  +  ^2  ^_  (,2  ^  2  «6  -  2  ac  -  2  be. 

Solve  the  following  equations  for  x : 

13.  (x  —  a){x  —  a)  =  —  x{a  —  X) -j- {a -^  b)x. 

,  ^     a         X 

14.  -  = 


b      c  —  x 


^^     4a,  4  a      3,5a 
3a;       X       2      ox 

2  a -3b      b      3a^l3a 
3a;  a;      2a;        6 

17.  {x  -  af  +  {x  -  by  =  2(x  -  cf. 

18.  ax  —  a(b  —  ic)  +  ac  =  3  ab. 

a;4-6       2x-{-a  x  —  b     x  —  a     x—  c 

20.    -A_  +  ^  =  .^.  22.       7  3  4 


a;  +  a     a;— 6      a;  —  a  a;  —  a     a;H-a     x-\-b 

23.  Solve  for  each  letter  in  terms  of  the  others : 

a      b      c 

24.  Solve  for  a        1  +  1  +  ^  +  ^  =  0. 

abed 

25.  The  sum  of  two  numbers  is  s  and  one  of   them  is   x. 
How  do  you  represent  the  other  ? 

26.  The  difference  between  two  numbers  is  d  and  one  of 
them  is  x.     How  do  you  represent  the  other  ? 

27.  The  sum  of  two  numbers  is  a  and  their  difference  is  b. 
Find  the  numbers  in  terms  of  a  and  b. 

28.  The  difference  between  two  numbers   is  a  and    one  is 
twice  the  other.     Find  each  number  in  terms  of  a. 


332 


REVIEW  EXERCISES 


REVIEW  EXERCISES  FOR   CHAPTER  XIV 


Solve : 


1. 


2. 


6. 


5x  +  Sy  =  1. 

x-y  =  S7, 

2x-^3y  =  SU-\-lSy. 

2x  —  Sy  =  y+6y 
x-2y  =  4.y  +  S. 

5x-{-3  y  =  0, 
2x-^y  =  l. 

2x  +  Sy  =  6x-lj 
3x-2y  =  3, 


10. 


11. 


[g  +  T      45-1^1 
3  6  2' 

4a  +  7      76+3^     ^ 
3  5 

■a;  +  2?/-}-22;  =  3, 
3x  —  4:y-\-z  =  19, 
-2x  +  ey-{-3z  =  0. 

2x-\-5y  -\-7 z  =  7, 
3x-9y-2z  =  23, 
-x-\-3y-\-3z=-10. 


5x-3y  =  0, 

3     4^6 

2x-\-2-6y  =  2-x. 
\5x-3      2y-5      1 

12. 

^  +  ^-A  =  3, 
2      8      12 

4                2       ~2' 
3a;  +  5      2/-10_g      • 

6     2     3 

[25 

a;  -(-  2/  -  2  =  35, 

r2a;-9     42/-2         ^ 
3              5 

13.     < 

^'  +  ^4-.  =  15, 
3^5 

2              4 

5     5 

14.  If  r  represents  the  rate  in  miles  per  hour  at  which  a 
train  is  moving,  how  far  will  it  go  in  t  hours  ?  Another  train 
runs  10  miles  per  hour  faster.  Express  in  symbols  the  sum  of 
the  distances  which  these  two  trains  travel  in  t  hours. 

15.  If  i\  is  the  rate  of  a  current  and  rg  the  rate  of  a  steamer 
in  still  water,  express  the  distance  which  the  steamer  can  go 
in  t  hours  :   (a)  with  the  current ;    (h)  against  the  current. 


REVIEW   EXERCISES 


333 


REVIEW  EXERCISES  FOR  CHAPTER  XV 

1.  Describe  the  method  of  locating  a  point  geographically 
on  the  earth's  surface. 

2.  Describe  the  method  for  locating  a  point  graphically  on 
squared  paper, 

3.  What  is  meant  by  a  linear  equation  in  two  unknowns  ? 
Write  such  an  equation. 

4.  How  many  points  do  you  need  to  locate  on  the  graph  of 
a  linear  equation  before  you  can  draw  the  whole  graph  ? 

5.  Construct  graphs  of  the  equations  3  ?/  —  2  ic  =  12  and 

6.  What  is  tlie  relation  between  the  graphs  of  the  equations 
a-  +  3  ?/  =  -  G  and  2  a;  -  4  2/  =  -  12  ? 

7.  What  is  meant  b}''  independent  equations  ?  by  simulta- 
neous equations  ? 

8.  Write  a  pair  of  equations  which  are  not  independent. 
Construct  their  graphs. 

9.  Write  a  pair  of  equations  which  are  not  simultaneous 
Construct  their  graphs. 

Solve  the  following  equations : 
5. 


10. 


11. 


12. 


13. 


14. 


♦C/  I  "It*/ 


a;  +  3 


x-2 

X  -\-  a 

a  -\-h  '  a  —  b 
\x^2y  =  4, 
\2x  +  y  =  -l. 
J  5  .X-  +  9  ?/  =  19, 
I  3  X  —  ?/  =  5. 

3x-7y  =  -ll, 

2.x-  +  y  =  4. 


15. 


16. 


17. 


5.T  — 3?/  =  4  — 2a;H-7?/, 
5y  -^  x=T. 
X  —  o  y  -\-  z  =  10, 
2x-\-y-z=l, 
'Sx-2y-{-5z  =  Sl. 
x-\-y-{-z  =  l, 
Sx-^4y  —  z  =  1, 
—  2x  —  y-\-3z  =  5, 


334  REVIEW   EXERCISES 

REVIEW   EXERCISES  FOR  CHAPTER  XVI 

Find  the  square  roots  of : 

1.  a;6  +  2 x'  -\-2x^  -{-x^  +  2x-j-l. 

2.  a;8  +  2.T'5  +  3a;4  +  2a;2  4_l. 

3.  x'-^4.x^  +  10  x'  4- 16  x^  +  17  x"'  +  12  a;  +  4. 

4.  4a^-12a3-7a2  +  24a  +  16. 
'         ,  ,  2a^2a^2  ^1^1 

0  C  DC        0^        G- 

4  g^     4  g^     13  g-     8  a     16 
'     6^         6^        3  6-'  "^  3  6       9  * 

O'    -7  i •  ~r  — r  — 

y^       z       z^       X       x^       y 

Approximate  the  square  roots  of  each  of  the  following  to 
two  places  of  decimals  : 

9.    7.9482.  11.   390.07.  13.   .0048. 

10.    4578.9.  12.    9.176.  14.    .04791. 

Simplify  the  following : 

15.    VT8;  V2O;  V^;  V^^x.     16.    ^|;  ^?;  ^|  ^^. 

17.  V45+VX;    V48+V75-V3. 

18.  Divide  x^  +  x*y  +  x^y^  -\-x^y^  -\-  xy*  -\-  y^  by  x  +  y. 
19     Divide  x^  —  y^  by  x"^  +  xy  +  2/"- 

20.  J^ivide  ^  —  7/  by  a.*^  +  x-y  +  iiv/^  +  y^. 

21.  The  older  of  two  sisters  is  now  8  years  less  than  twice  as 
old  as  the  other.  If  x  represents  the  age  of  the  younger  sister, 
represent  in  symbols  twice  the  sum  of  their  ages  7  years  ago. 

22.  A  rear  wheel  of  a  wagon  has  a  circumference  4  feet 
greater  than  that  of  a  front  wheel.  If  tlie  circumference  of 
the  rear  wheel  is  x  feet,  represent  in  symbols  the  number  of 
revolutions  each  wheel  must  make  to  go  one  mile. 


REVIEW   EXERCISES  335 

REVIEW   EXERCISES  FOR  CHAPTER  XVII 

Simplify  the  following : 


3.    va^  -f-  2  c^b  +  a62  +  Va^  -  2  a^ft  -f  a^^ _  2 Va^ 


4.    Va3  -  a25  _  V^^a  -  b^  +  Va^^^  -  a''b\ 


5.  V(.y  +  ?/)(aJ^  -  2/^)  -  V(a;  +  yy{x  -  y)—  Vx^  -  d;^^. 
Solve  the  following  equations  : 

6.  Va;2  -  8  +  a;  =  8.  7.    Vo a  -  24  +  -i  =  V^a 


8.    •\/2a-l=7-V2a+6. 


9.    V.i-  +  2  =  Va;  —  6  +  2 Va;  —  5. 

V5  g;  +  1  ^     /9  a;  4-1 


10.    V2^^-2  =  a;+l.  13. 

11 


V2a.--5       ^3a;-o 
V2a;  +  7=  Va;  +  2.  ^4     V.^M^  =  3  -  V3^^. 


12. 


Va;-a         2v'a  V8a;  +  1 


*  Rationalize  the  denominators  of  the  following : 
Va  —  Va;  a  —  V6 


_  „     -y/a  4-  a;  +  Va  —  a;  -  „     Va;  —  a  +  & 

Va  +  a;  —  V«  —  x  Va;  4-  ot  —  c 

20.  — ? 22.        "  +  ^      .  24.    ^^l+I. 

V7-V4  Va  +  V6  V8-7 

21.  V1±V2.  ^3     24:V3.  ,3     iV2-_l. 
V7-V2                      4-V3  iV2  +  l 

26.  A  and  B  working  together  can  do  a  piece  of  work  in  12 
days.  B  and  C  working  together  can  do  it  in  13  days,  and  A 
and  C  working  together  can  do  it  in  10  days.  How  long  will 
it  require  each  to  do  it  when  working  alone? 


336  REVIEW   EXERCISES 

REVIEW  EXERCISES  FOR  CHAPTER  XVIII 

Complete  the  square  in  each  of  the  following : 

1.  4  a-  +  8  a.  4.   25  oi^  -  7  x.  7.   3  x^  -4  a;. 

2.  9  a^  +  30  a,  5.    7  h^  -  3  b.  8.    7  x^  -  11  x. 

3.  0^2  ^  3  ^.^  6     16  ^>2  _  7  5.  9.    8  a;2  4-  7  x. 

Solve  and  check,  using  §  204 : 

10.  a;2  -  7  ic  +  9  ■-=  0.  17.    4  aV  +  6  6a;  =  3  61 

11.  2a-2-5a^  +  2  =  0. 

12.  7  a;2  +  18  a;  -  3  =  0. 

13.  a:2  -  12  a;  +  16  =  0. 

14.  a;2  +  2  6a;  =  3  c. 

15.  2  a;^  —  5  aa;  =  a^ 

16.  aa;2  H-  2  6a;  =  3  c. 

Reduce  the  solution  of  each  of  the  following  to  simplest 
form  without  approximating  any  roots : 

24.  7  a;2  =  27.  30.    7  ax^  =  98. 

25.  5a;2  =  108.  31.    2  (a  +  6)  .'r^  =  300. 

26.  2a.'2  =  3.  32.    x'=-^^, 

125  ab 

27.  3a2  =  343.  .^^ 

33.    ar^  = 


18. 

a;2  _  18  a;  +  4  =  0. 

19. 

a;2  —  3  aa;  +  6  =  0. 

20. 

a;2  +  9  6a;  4-  c  =  0. 

21. 

2  a;2  -  7  a;  =  5. 

22. 

3  ax''  -  7  hx  =  3. 

23. 

4  a'x'  —  2  6aa;  =  ca. 

28.  5  ar  =  8  (X^6.  '    '         72  cc?^* 

29.  2  ar  =  27  a(a  +  6)^.  34.    (a  —  6).r  =  a  +  6. 

35.  The  circumference  of  the  rear  wheel  of  a  carriage  is  1.8 
feet  more  than  that  of  the  front  wheel.  In  running  one  mile 
the  front  wheel  makes  48  revolutions  more  than  the  rear 
wheel.     Find  the  circumference  of  each  wheel. 

36.  The  circumference  of  the  rear  wheel  of  a  carriage  is  1 
foot  more  than  that  of  the  front  wheel.  In  going  one  mile  the 
two  wheels  together  make  920  revolutions.  Find  the  circum- 
ference of  each. 


REVIEW  EXERCISES  337 

REVIEW  EXERCISES  FOR   CHAPTER  XIX 

Solve  the  following  systems  of  equations  : 

(  x2  -f-  2/2  =  25,  [2  x^  -37f  =  7, 

1.  6. 

I  x  —  2/  =  4.  \^x  —  y  =  5. 

I2x^-Sxy  =  12,  ^  [a^_7/  =  56, 

[x  +  2y  =  4..  [x-y  =  2. 

\x-2y  =  3. 

^      {xy-f=U,  ^^ 

'     \^x-\-y  =  4:.  '      [  a:^  —  4  a;?/ +  4  2/2  =  0. 

\x'-xy  =  S,  ^^  ^   (  X'  -  xy  =  3, 

\x-^y  =  -2.  '     \y'-{-Sxy  =  22. 

11.  Divide  x'-  3  x*-  18  a.-^  +  24  x''-\-  52  .r  -  21  by  x''+  x  -  7. 

12.  Divide  6  a5  +  5  a4-60a3-|_4  a^  +  Tl  a+28  by  3  a2-5  a-4. 

Find  the  square  roots  of : 

13.  16  x^  -  40  xy  +  xY  +  30  xy  +  9  y^\ 

14.  4  m^  —  20  m^  +  9  ??i*  +  52  m^  —  14  7/1^  —  24  m  -4-  9. 

15.  Two  square  pieces  of  land  require  together  360  rods  of 
fence.  If  the  difference  in  the  area  of  the  pieces  is  900  square 
rods,  how  large  is  each  piece  ? 

16.*  The  difference  of  the  cubes  of  two  numbers  is  218  and 
the  difference  of  the  numbers  is  2.     Find  the  numbers. 

17.  The  hind  wheel  of  a  wagon  makes  12  revolutions  less 
than  the  fore  wheel  in  going  720  feet.  If  the  circumference 
of  each  wheel  were  three  feet  greater,  the  hind  wheel  would 
make  8  revolutions  less  than  the  fore  wheel  in  going  the  same 
distance  as  before.     Find  the  circumference  of  each  wheel. 


338  REVIEW   EXERCISES 

REVIEW  EXERCISES   FOR   CHAPTER  XX 

Expand  by  the  binomial  formula : 
1.    iSx-2yy.  4.    (2^  +  3^).* 

7.  Find  the  6th  term  of  {x  +  y)". 

8.  Find  the  8th  term  of  (2  a  -  b)^. 

9.  Find  the  7th  term  of  (a  —  3  by. 

10.  Find  the  13th  term  of  (2  a  —  c)". 

11.  What   is   the    value   of   x^  —  S  x   if   x=1—Vd?       If 
a-  =  1  -  ^  5  ? 

12.  What  is  the  value  of  3  .t-^  -  5  a-  +  6  if  .r  =  ^—^^  ?     If 
2+^3o  - 


X  = 


J*  = 


13.    What   is   the   value   of   ox'  +  Tx  if   x  = ''  ^^  "  ?      If 


Find  the  square  root  of  each  of  the  folloTving : 

14.  2oa2  +  c2  +  962-10ao  +  30a6-66c. 

15.  a^+y--r^ '"4-4  >-—  2 x[i  -\-  \ xz  —  4 xi*  —  4  j/z  -f-  4 yi*  —  8 zr. 

16.  x^  —  'Ix"  —  r  -roxr  -\-2x  ^\. 

17.  .r*- 6a^  +  13.i--t2.r-4. 

18.  Divide  6j:^  +  j^-f  12^4-8  bv  2j^-j--h 2. 

19.  Divide 3x«+4r5-jr*+6j:^-12j^ -1-8 j--12by3jc2-2ar+3. 

20.  Divide  3  a"  —  5  a'  -t-  8  a'  4-  2  a=  -  18  a  -h  12  by  o^  —  a  -h  2. 

21.  Divide  6  a^  -h  10  a'  -  9  a-  +  11  a  -  6  by  2  a^  -h  4  a  -  3. 

22.  A  picture  inside  the  frame  is  !r  inches  ^vide  and  \  inches 
long.  The  frame  is  a  inches  wide.  Express  the  area  of  the 
frame  in  terms  of  a,  ir,  and  I. 


REVIEW   EXERCISES  339 

REVIEW  EXERCISES   FOR   CHAPTER   XXI 

1.  It  y OCX  and  if  ?/  =  9  when  .i"=17,  lincl  y  when  x=  12. 

2.  If  y  =  -  and  if  ?/  =  4  when  x  =  ^,  find  y  when  ic  =  f . 

3.  If  yccx-  and  if  ?/  =  7  when  x  =  1,  find  y  when  x  =  4. 

4.  If  y  ccx^  and  if  y  =  3  when  x  =  3,  find  7  when  x  =  T. 

5.  If  2/  QC  ic^  and  if  ?/  ==  G  when  .r  =  4,  find  y  when  .'c  =  10. 

6.  If  a  certain  sum  of  money  yields  S  350  interest  in 
8  months,  how  much  will  the  same  sum  yield  in  17|-  months  ? 

7.  If  $  2500  yields  $  680  interest  during  a  certain  time, 
how  much  money  will  be  required  to  yield  $390  during  the 
same  time  ? 

8.  If  60  men  can  do  a  certain  piece  of  work  in  45  days, 
how  long  will  it  take  35  men  to  do  it  ? 

9.  Over  what  distance  will  a  falling  body  pass  in  5  seconds, 
starting  from  rest? 

10.  Over  w^hat  distance  will  a  body  fall  in  10  seconds,  if  it 
starts  with  a  velocity  of  15  feet  per  second  ? 

11.  If  a  circular  disk  weighs  16  lbs.,  how  much  will  a  disk 
weigh  if  its  diameter  is  3  times  that  of  the  first  disk  and  if  it 
is  of  the  same  material  and  thickness  ? 

1.1,1  a  a 


ah      etc      be  b'          (b  —  aY 

22.     .  14, |- -^ —> 

(a—  b  —  c )  (a-\-  b  -t  c)  '    -1    .  ^        6  +  a 

ab  b"^       b  —  a 

a-5Y  +  16  2  +  -J— 4-a^ 

a  X  —  2 

13.    ^^ ^^r^ 15 


a  +  -)+4  x  +  -^-2 


aJ  x  +  2 


340 


REVIEW  EXERCISES 


MISCELLANEOUS  REVIEW  EXERCISES 

Solve  the  following : 

1.  (x-2y-(x-l){x  +  2)  =  6-5x. 

2.  {X  -  2){x  +  2)  +  (3  ;^  -  1)(2  -  x)  =  (X  -  2)(5  -  2  x). 

3.  ^x-3y+{2x-j-5y=(5x-3)(x-}-5)-7. 


(2  _  xy  -{2x-iy  =  (-Sx  +  1)(4  +  x)  -4. 


6. 


7. 


8. 


{2x-3y-l  =  0, 

\5x  +  2y  =  12, 

lx  +  y  =  a, 
\x-y  =  b. 

{  ax-\-by  =  Cj 
[  ax  —  by  =  d, 

J  a     0 
\  X      y       -J 
{ah 


10. 


11. 


12. 


3ax  —  hy  =  2, 
[2x  +  3hy  =  Q. 
x  —  y  —  S  z=.—  6, 
•  2x-j-y  —  z  =  ll, 
^  —x-i-3y  -{-z  =  16. 
\  2x  —  3by  =  c, 
2ax  —  5y  =  d. 
2x-y  +  3z  =  20, 
x  +  4:y-z  =  -2, 
5x  +  y  —  6z  =  6. 

13.  {x-iy-(x-S){2x-l)  =  -x''  +  9S. 

14.  (7  +  x){x-4)-{-(l-xy  =  -23-\-2x\ 

15.  (12  -  4  x)(2  -  a;)-  4(1  +  xy  =  5  a;  +  119. 

16.  {x-  17)  (59  -2x)-(l-xy=(6-3  x){x  -  2)  +  384. 

17.  (3x-2)  +  (x-iy+  {x-2y=2(^x-l){x-2)-h5. 

18.  (G  -  3  x)(2  +x)-\-  W(x  -  1)2  =  13(x  +  4)2  +  364. 

19.  7a;4-(8a;  +  4)-T-2  =:4rc  +  9. 

20.  Ga;  + 4(4.^  +  2)  =85-3(2.^  +  7). 

21.  8  +  7(6  +  6  7i)+  2  n  =  2(4  n  +  5)+  18n  +  49. 

22.  5(9  a;  +  3)  +  6  x  =  24  x  -  4(3  x  -\-  2)  +  36. 

23.  Find  the  average  of  the  following  temperatures  :  7  a.m., 
-  4° ;  8  A.M.,  -  2° ;  9  a.m.,  -  1°;  10  a.m.,  +  1° ;  11  a.m.,  +  5° ; 
12  m.,  +  7°. 


REVIEW  EXERCISES 


341 


Solve  the  following: 

2  aa;  +  2  2/  =  4. 

6  a;  +  3  2/  =  1, 
5ax  —  2by  =  c. 


24 


25 


26. 


27. 


I(12^±8)  +  i3  +  5^-6  =  47. 
4 


a^  4- 2  _  X  -  1      3  ic  +  2 


a 


28. 


30. 


32. 


'  7  a;  —  3      a^/  +  4  _  -i 
2  3      ~    ^ 

6a;  +  5      y  —  ^  ^  g 
.43 

'2x  —  y  —  z  =  S, 
3  a;  +  2  2/  +  ;2  =  24, 
[  —a;  — 3^  +  52;  =  16. 

6  4c 


6 

f2  5.v 


29.     { 


3  a 


ab 


3  a:  =  2, 


5  2/  +  — —  =  5. 


a 


f8aj-32/-2  =  0. 
31.    I  2  a; +  2  2/ 4- 3:2  =  10, 
[-x  +  y  +  6z  =  S. 

a  —  b 


2x-\      (2a;  +  l)(2aj-l)      2a;  +  l 

33.  (1-3  xY  +  (2  x  +  1)2  =  5  .^2  +  (2  a;  H-  6)(4  x  +  27). 

34.  (14-2  a;)  (2-3  a;)  +  (a; -4) (a; -4-4)  =  (a; 4- 14) (18 -5  a;)  -  1. 

35.  A  picture  inside  the  frame  is  12  inches  long  and  8 
inches  wide.  If  the  frame  is  a  inches  wide,  express  its  area  in 
terms  of  a. 

36.  If  li  is  the  length  of  the  hypotenuse  of  a  right  triangle 
and  a  the  length  of  one  side,  express  the  length  of  the  third 
side  in  terms  of  li  and  a. 

37.  A  rectangular  piece  of  tin  is  to  inches  wide  and  I  inches 
long.  If  a  square  a  inches  on  a  side  is  cut  out  of  each  corner, 
express  in  terms  of  w,  /,  and  a  the  volume  of  a  box  formed  by 
turning  up  the  sides. 

38.  A  farmer  plows  a  strip  a  rods  wide  around  a  rectangular 
tield  w  rods  wide  and  I  rods  long.  Express  in  terms  of  w^  I, 
and  a  the  area  plowed. 


342 


REVIEW   EXERCISES 


Solve  the  following : 

39.    2{x  +  o){x  -  5)  =  {x  -  b){x  +  1)  +  (a;  -  2)^  -  1. 


40. 


41. 


42. 


43. 


44. 


45. 


ix-l 
2 

3 

3i, 

.^•4- 1 
3 

4 

.  7 
■  6* 

X  —  a 
2 

1  .y  -  ^  _ 
3 

=  1, 

X  -\-  a 
3 

■^     2 

1. 

46.      { 


(  X  -\-  ay  =  c, 


47. 


2  ax  —  3  by  =  c, 
2  ax  -{-  3  by  =  d. 

{x  +  y  -]-z  =  6, 
I2x  —  y-\-z  =  3, 
[3x-\-2y-2z  =  l. 

f  X  —  y  -\-  z  =  i, 
\x-2y-{-4.z  =  6, 
[2x-{-y-3z  =  10. 
f  X  -{-  y  -{-  z  =  0, 
\2x-4.y  +  z=-3, 
[3x-\-2y-}-4:Z  =  3. 


[  bx  -\-y  =  d. 
ax  -\-  3  y  =  2  c, 
bx  —  2  y  =  3  d. 

X      y 
48.     {^  +  ^  =  b, 

y    ^ 
1  ,  1 

-  +  -  =  c. 

12         it- 


49. 


50. 


51. 


52. 


f  aa;  4-  5?/  =  1, 

CO,'  +  dy  =  4. 

aic  —  by  =  c, 
[  ex  +  f?//  =  e. 
(  X  -\-  y  -^  z  =  a, 

2x-2y  +  2z  =  b, 
\3  X  —  y  —  z  =  c. 
^  ax  —  y  +  bz  =  a, 

X  -f  ay  —  z  =  1, 

bx  —  y  +  az  =  b. 


53.  If  vj  and  I  are  the  length  and  width  of  a  rectangle, 
express  in  symbols  the  length  of  its  diagonal. 

54.  The  lengths  of  the  two  sides  of  a  right  triangle  are  !(> 
and  24  respectively.  Express  the  length  of  the  hypotenuse 
in  the  simplest  form  without  approximating  a  square  root. 

55.  A  father  is  now  twice  the  age  of  his  son.  If  x  repre- 
sents the  son's  age  now,  express  twice  the  sum  of  their  ages 
5  years  ago. 

56.  One  number  is  3  less  than  4  times  another.  Express 
one  third  the  sum  of  the  numbers  if  x  represents  one  of  them. 


REVIEW   EXERCISES  343 

57.  If  h  is  the  digit  in  hundreds'  place,  t  the  digit  in  tens' 
place,  and  u  the  digit  in  units'  place  of  a  certain  number, 
express  the  number  obtained  by  inverting  the  order  of  the 
digits  of  the  given  number. 

58.  ^,  tj  and  u  are  the  digits  in  hundreds',  tens',  and  units' 
places  of  a  number.  Express  the  number  obtained  by  increas- 
ing each  digit  by  2. 

^3     12(5  +  4a.)_5(6  +  4a:)      ^^^  3^ 

6  2 

60.    15  I  21(3  +  0^)   ^  2(6  +  18  x)^  3(9  0^  +  12)  ^  ^g 
7  3  3 

ll(5a;  +  25)      3(6  a;  -  2)  ^  7(4  a;  +  8)      12  a;  +  36      3^ 
5  2  4  3 


62. 
63. 


2(a;  +  1)     3  ^      a.-  -  1 

a;  —  1  a?  4- 1 

64.    \ 

X  +  a  _  ^  _  4(a;  —  a)  I  3  a  +  6  _  5  —  46  _  r. 


[  2a  +  5      36-10  ^  ^ 
3  4 


a;  —  a  ;c  +  a  18  7 


PROBLEMS   ON   MOTION 

1.  A  sparrow  flies  135  feet  per  second  and  a  hawk  149  feet 
per  Second.  The  hawk  in  pursuing  the  sparrow  passes  a  cer- 
tain point  7  seconds  after  the  sparrow.  In  how  many  seconds 
from  this  time  does  the  hawk  overtake  the  sparrow  ? 

2.  A  courier  starts  from  a  certain  point,  traveling  r^  miles 
per  hour,  and  a  hours  later  a  second  courier  starts,  going  at 
the  rate  of  ra  miles  per  hour.  In  how  long  a  time  will  the 
second  overtake  the  first,  supposing  i\  to  be  greater  than  1\  ? 

If  the  second  courier  requires  t  hours  to  overtake  the  first,  the  latter 
had  been  on  the  way  t-\-  a  hours.  Thus  the  distance  covered  by  the 
second  courier  is  r^t  and  by  the  first  ri{t  +  a).  As  these  numbers  are 
equal  we  have  r^  =  r,{t  +  a). 

This  formula  summarizes  the  solution  of  all  problems  like  3  and  4 
on  the  next  page. 


i 


344  REVIEW   EXERCISES 

3.  In  an  automobile  race  A  drives  his  machine  at  an 
average  rate  of  53  miles  per  hour,  while  B,  who  starts  J  hour 
later,  averages  57  miles  per  hour.  How  long  does  it  require  B 
to  overtake  A  ?     Use  the  formula  on  page  343. 

4.  A  freight  steamer  leaves  New  York  for  Liverpool,  aver- 
aging 10^  knots  per  hour,  and  is  followed  4  days  later  by  an 
ocean  greyhound,  averaging  25^  knots  per  hour.  In  how  long 
a  time  will  the  latter  overtake  the  former  ? 

5.  One  athlete  makes  a  lap  on  an  oval  track  in  26  seconds, 
another  in  28  seconds.  If  they  start  together  in  the  same 
direction,  how  soon  will  the  first  gain  one  lap  ?     Two  laps  ? 

Let  one  lap  be  the  unit  of  distance.  Since  the  first  covers  one  lap 
in  26  seconds,  his  rate  per  second  is  ■^^.  Likewise  the  rate  of  the  other 
is  2^-  If  t  is  the  required  number  of  seconds,  the  distance  covered  by 
the  first  is  j^g  t  and  by  the  second  -^^t.  If  the  first  goes  one  lap  farther 
than  the  second,  the  equation  in  ^t  =  ^^t  +  \. 

6.  Two  automobiles  are  racing  on  a  circular  track.  One 
makes  the  circuit  in  31  minutes  and  the  other  in  38^  minutes. 
In  what  time  will  the  faster  machine  gain  1  lap  on  the  slower? 

a  7.    The   planet    Mercury    makes    a 

circuit  around  the  sun  in  3  months 
and  Venus  in  74  months.  Startinsr  in 
Venua  conjunction,  as  in  the  figure,  how  long 
before  they  will  again  be  in  this  posi- 
tion ? 

Note  that  the  problem  may  be  solved  just 
as  if  the  two  planets  were  moving  in  the  same  orbit  at  different  rates. 

8.  Saturn  goes  around  the  sun  in  29  years  and  Jupiter  in 
12  years.     Find  the  time  between  two  conjunctions. 

9.  Uranus  makes  the  circuit  of  its  orbit  in  84  years  and 
Neptune  in  164  years.  If  they  start  in  conjunction,  how  long 
before  they  will  be  in  conjunction  again?     Aiis.  172^  years. 


REVIEW    EXERCISES  345 

10.  The  hour  hand  of  a  watch  makes  one  revolution  in  12 
hours  and  the  minute  hand  in  1  hour.  How  long  is  it  from 
the  time  when  the  hands  are  together  until  they  are  again 
together  ? 

11.  One  object  makes  a  complete  circuit  in  a  units  of  time 
and  another  in  b  units  (of  the  same  kind).  In  how  many  units 
of  time  will  one  overtake  the  other,  supposing  b  to  be  greater 
than  a  ? 

The  solution  of  this  problem  summarizes  the  solution  of  all  problems 
like  those  from  5  to  10. 

12.  At  what  times  between  12  o'clock  and  6  o'clock  are  the 
hands  of  a  watch  together  ?  (Find  the  time  required  to  gain 
one  circuit,  two  circuits,  etc.) 

PROBLEMS  INVOLVIWG  THE  LEVER 

1.  A  teeter  board  is  in  balance  when  two  boys,  A  and  B, 
weighing  105  and  75  pounds  respectively,  are  seated  at  dis- 
tances 5  and  7  feet  from  the  fulcrum,  because  7  •  75  =  5  •  105. 
If  now  two  boys  weighing  48  and  64  pounds  are  seated  on  the 
same  board  Avith  the  other  boys,  the  teeter  will  again  be  in 
balance  if  their  distances  are  4  and  3  feet,  because 
7  .  75  -h  4  .  48  =  5  .  105  +  3  •  64 

g(l05  lbs.;    Z)(64  lbs.; C'(48  lbs.)      (75  lbs.)  J 

I  . 3-feet A: 4-feet ] 

6  feet  ^^      '~        ~         ~  7  feet 

The  weight  of  the  boy  multiplied  by  his  distance  from  the 
fulcrum  is  called  his  leverage.  The  sum  of  the  leverages  on 
the  two  sides  must  be  the  same.  Hence,  if  the  teeter  balances 
when  two  boys,  weighing  respectively  ii\  and  iVo  pounds,  are  at 
distances  di  and  do  on  one  side,  and  two  boys,  weighing  w^  and 
w^  pounds,  are  at  distances  d^,  d^  on  the  other  side,  then 


346  REVIEW   EXERCISES 

2.  If  two  boys  weighing  75  and  90  pounds  sit  at  distances 
of  3  and  5  feet  respectively  on  one  side  and  one  weighing 
82  pounds  sits  at  3  feet  on  the  other  side,  where  should 
a  boy  weighing  100  pounds  sit  to  make  the  board  balance  ? 

3.  A  beam  carries  a  weight  of  240  pounds  7^  feet  from  the 
fulcrum  and  a  weight  of  265  pounds  at  the  opposite  end  which 
is  10  feet  from  the  fulcrum.  On  which  side  and  how  far  from 
the  fulcrum  should  a  weight  of  170  pounds  be  placed  so  as  to 
make  the  beam  balance  ? 

4.  Two  boys,  A  and  B,  having  a  50-lb.  weight  and  a  teeter 
board,  proceed  to  determine  their  respective  weights  as  fol- 
lows :  They  find  that  they  balance  when  B  is  9  feet  and  A  is 
7  feet  from  the  fulcrum.  If  B  places  the  50-lb.  weight  on  the 
board  beside  him,  they  balance  when  B  is  3  and  A  is  4  feet 
from  the  fulcrum.     How  heavy  is  each  boy  ? 

5.  C  is  6i  feet  from  the  point  of  support  and  balances  D, 
who  is  at  an  unknown  distance  from  this  point.  C  places  a 
33-lb.  weight  beside  himself  on  the  board  and,  when  4|  feet  from 
the  fulcrum,  balances  I),  who  remains  at  the  same  point  as  be- 
fore. D's  weight  is  84  pounds.  What  is  C's  weight,  and  how 
far  is  D  from  the  fulcrum  ? 

6.  E  weighs  95  pounds  and  F  110  pounds.  They  balance 
at  certain  unknown  distances  from  the  fulcrum.  E  then  takes 
a  30-pound  weight  on  the  board,  which  compels  F  to  move  3  feet 
farther  from  the  fulcrum.  How  far  from  the  fulcrum  was  each 
of  the  boys  at  first  ? 

7.  A  who  weighs  60  pounds  sits  3  feet  from  the  fulcrum. 
B  who  weiglis  90  pounds  sits  4  feet  from  the  fulcrum  on  the 
same  side  as  A.  How  far  from  tlie  fulcrum  on  the  opposite 
side  must  C,  who  weighs  108  pounds,  sit  in  order  to  make  the 
teeter  board  balance  ? 


REVIEW   EXERCISES 


347 


PROBLEMS   INVOLVING    GEOMETRY 

1.  A  picture  is  4  inches  longer  than  it  is  wide.  Another 
picture,  which  is  12  inches  longer  and  6  inches  narrower,  con- 
tains the  same  number  of  square  inches.  Find  the  dimensions 
of  the  pictures. 

2.  A  picture,  not  including  the  frame, 
is  8  inches  longer  than  it  is  wide.  The 
area  of  the  frame,  which  is  2  inches  wide, 
is  176  square  inches.  Find  the  dimensions 
of  the  picture. 


E222S2S^SS^222?1 


x+8 


7i;>;i;:;>;i''->i>;f';i'r!.':i'';iii;f;,'// 


3.  A  picture,  including  the  frame,  is  10  inches  longer  than 
it  is  wide.  The  area  of  the  frame,  which  is  3  inches  wide,  is 
192  square  inches.     What  are  the  dimensions  of  the  picture? 

4.  The  base  of  a  triangle  is  11  inches  greater  than  its  alti- 
tude. If  the  altitude  and  the  base  are  both  decreased  7  inches, 
the  area  is  decreased  119  square  inches.  Find  the  base  and 
the  altitude  of  the  triangle. 

5.  The  base  of  a  triangle  is  3  inches  less  than  its  altitude. 
If  the  altitude  and  the  base  are  both  increased  by  5  inches, 
the  area  is  increased  by  155  square  inches.  Find  the  base 
and  the  altitude  of  the  triangle. 

6.  A  square  is  inscribed  in  a  circle  and  another  circum- 
scribed about  it.  The  area  of  the  strip  inclosed  by  the  two 
squares  is  25  square  inches.     Find  the  radius  of  the  circle. 

7.  Find  the  sum  of  the  areas  of  a  circle  of  radius  6  and  the 
square  circumscribed  about  the  circle. 

The  area  of  the  circle  is  6-  tt  =  36  tt,  and  the  area 
of  the  square  4  •  6-  =  4  •  36  ;  i.e.  the  square  contains  4 
squares  whose  sides  are  6.     The  sum  of  the  areas  is 

4  .  36  +  36  TT  =  (4  +  7r)36  =  (4  +  3})  36. 


348 


REVIEW   EXERCISKS 


8.  Find  an  expression  for  the  sum  of  the  areas  of  a  circle 
of  radius  r  and  the  circumscribed  square.  (Solve  Example  7 
by  substituting  in  the  formula  here  obtained.) 

9.  If  the  sum  of  the  areas  of  a  circle  and  the  circumscribed 
square  is  64,  find  the  radius  of  the  circle. 

By  the  formula  obtained  under  Example  8, 
64  =  (4^^)  r-^  =  A.0  r2. 
Hence,  r  =  V  8^  =  2.99. 

10.  If  the  sum  of  the  areas  of  a  circle  and  the  circumscribed 
square  is  640  square  feet,  find  the  radius  of  the  circle. 

11.  The  sum  of  the  areas  of  a  circle  and  the  circumscribed 
square  is  a.  Find  an  expression  representing  the  radius  of 
the  circle.     (Replace  tt  by  3i  before  simplifying.) 

12.  If  the  radius  of  a  circle  is  12,  j5nd  the 
difference  between  the  areas  of  the  circle  and 
the  circumscribed  square. 

13.  If  the  radius  of  a  circle  is  r,  find  the  dif- 
ference between  the  areas  of  the  circle  and  the 
circumscribed  square.     (Solve  Example  12  by 

the  use  of  the  formula  obtained  here.) 

14.  If  the  radius  of  a  circle  is  16,  find  the  area  of  the 
inscribed  square.  (This  is  the  same  problem  as  finding  the 
area  of  a  square  whose  diagonal  is  32.) 

15.  If  the  radius  of  a  circle  is  r,  find  an  expression  represent- 
ing the  area  of  the  inscribed  square. 

16.  If  the  radius  of  a  circle  is  12,  find  the 
difference  between  the  area  of  the  circle  and  the 
area  of  the  inscribed  square. 

17.  If  the  radius  of  a  circle  is  r,  find  an 
expression  representing  the  difference  be- 
tween the   areas   of   the  circle   and   of  the    inscribed   squara 


REVIEW   EXERCISES  349 

18.  The  radius  of  a  circle  is  10.     Find  the  area  of  an  in- 
scribed hexagon. 

19.  The  radius  of  a  circle  is  6.     Find  the  difference  between 
the  areas  of  the  circle  and  the  inscribed  hex- 
agon. 

20.  Find  an  expression  representing  the  dif- 
ference between  the  areas  of  a  circle  with  radius 
r  and  the  inscribed  regular  hexagon. 

MISCELLANEOUS  PROBLEMS 

1.  Divide  the  number  645  into  two  parts,  such  that  13 
times  the  first  part  is  20  more  than  6  times  the  other. 

2.  Divide  the  number  a  into  two  parts,  such  that  h  times 
the  first  part  is  c  more  than  d  times  the  second  part. 

3.  The  sum  of  three  numbers  is  98.  The  second  is  7 
greater  than  the  first,  and  the  third  is  9  greater  than  the  second. 
What  are  the  numbers  ? 

4.  The  sum  of  three  numbers  is  s.  The  second  is  a  greater 
than  the  first,  and  the  third  is  b  greater  than  the  second. 
What  are  the  numbers  ? 

5.  One  boy  runs  around  a  circular  track  in  26  seconds,  and 
another  in  30  seconds.  In  how  many  seconds  will  they  again 
be  together,  if  they  start  at  the  same  time  and  place  and  run 
in  the  same  direction  ? 

6.  A  bird  flying  with  the  wind  goes  65  miles  per  hour,  and 
flying  against  a  wind  twice  as  strong  it  goes  20  miles  per  hour. 
What  is  the  rate  of  the  wind  in  each  case  ? 

7.  A  steamer  going  with  the  tide  makes  19  miles  per  hour, 
and  going  against  a  current  4-  as  strong  it  makes  13  miles  per 
hour.     What  is  the  speed  of  the  steamer  in  still  water  ? 


850  REVIEW   EXERCISES 

8.  Find  the  time  between  4  and  5  o'clock  when  the  hand* 
of  the  clock  are  30  minute  spaces  apart. 

9.  A  man  takes  out  a  life  insurance  policy  for  which  he 
pays  in  a  single  payment.  Thirteen  years  later  he  dies  and  the 
company  pays  $  12,600  to  his  estate.  It  was  found  that  his 
investment  yielded  2  %  simple  interest.  How  much  did  he 
pay  for  the  policy  ? 

10.  After  deducting  a  commission  of  3  %  for  selling  bonds, 
a  broker  forwarded  $  824.50.  AVhat  was  the  selling  price  of 
the  bonds  ? 

11.  A  broker  sold  stocks  for  %  1728  and  remitted  $  1693.44 
to  his  employer.     What  was  the  rate  of  his  commission  ? 

12.  The  difference  between  the  areas  of  a  circle  and  its  cir- 
cumscribed square  is  12  square  inches.  Find  the  radius  of 
the  circle.     (See  problem  11,  page  348.) 

13.  The  difference  between  the  areas  of  a  circle  and  its  in- 
scribed square  is  12  square  inches.    Find  the  radius  of  the  circle. 

14.  The  difference  between  the  areas  of  a  circle  and  the 
regular  inscribed  hexagon  is  12  square  inches.  Find  the  radius 
of  the  circle. 

15.  The  altitude  of  an  equilateral  triangle  is  6.  Find  its  ^side 
and  also  its  area.     Find  the  side  and  area  if  the  altitude  is  li. 

16.  The  radius  of  a  circle  is  3  feet.  Find  the  area  of  the 
regular  circumscribed  hexagon.  Find  the  area  if  the  radius  is 
T  feet. 

17.  The  radius  of  a  circle  is  r.  Find  the  difference  between 
the  areas  of  the  circle  and  the  regular  circumscribed  hexagon. 

18.  The  difference  between  the  areas  of  a  circle  and  the 
regular  circumscribed  hexagon  is  9  square  inches.  Find  the 
radius  of  the  circle. 


REVIEW   EXERCISES  351 

19.  A  circle  is  inscribed  in  a  square  and  another  is  circum- 
scribed about  it.  The  area  of  the  ring  formed  by  the  two  cir- 
cles is  25  square  inches.     How  long  is  the  side  of  the  square  ? 

20.  In  a  building  there  are  at  work  18  carpenters,  7  plumbers, 
13  plasterers,  and  6  hod  carriers.  Each  plasterer  gets  $  1.90 
per  day  more  than  the  hod  carriers,  the  carpenters  get  35  cents 
per  day  more  than  the  plasterers,  and  the  plumbers  50  cents 
per  day  more  than  the  carpenters.  If  one  day's  wages  of  all 
the  men  amount  to  $  183.45,  how  much  does  each  get  per  day  ? 

21.  A  train  running  46  miles  per  hour  leaves  Chicago  for 
New  York  at  7  a.m.  Another  train  on  the  same  road  running 
56  miles  per  hour  leaves  at  9.30  a.m.  Find  when  the  trains 
will  be  15  miles  apart.     (Two  answers.) 

22.  There  is  a  number  consisting  of  three  digits,  those  in 
tens'  and  units'  places  being  the  same.  The  digit  in  hundreds' 
place  is  4  times  that  in  units'  place.  If  the  order  of  the  digits 
is  reversed,  the  number  is  decreased  by  594.  What  is  the 
number  ? 

23.  A  hound  pursuing  a  deer  gains  400  yards  in  25  minutes. 
If  the  deer  rune  1300  yards  a  minute,  how  fast  does  the  hound 
run  ?  If  the  hound  gains  v^  yards  in  t  minutes,  and  the  deer 
runs  V2  yards  per  minute,  find  the  speed  of  the  hound. 

24.  The  altitude  of  Popocatepetl  is  1716  feet  less  than  that 
of  Mt.  Logan,  and  the  altitude  of  Mt.  St.  Elias  is  316  feet 
greater  than  that  of  Popocatepetl.  Find  the  altitude  of  each 
mountain,  the  sum  of  their  altitudes  being  55,384  feet. 

25.  It  is  4  times  as  far  from  New  York  City  to  Cincinnati 
as  from  New  York  to  Baltimore.  Twice  the  distance  from 
New  York  to  Cincinnati  minus  5  times  that  from  New  York 
to  Baltimore  equals  567  miles.  How  far  is  it  from  New  York 
to  each  of  the  other  cities  ? 


352  REVIEW   EXERCISES 

26.  A  disabled  steamer  240  knots  from  port  is  making  only 
4  knots  an  hour.  By  wireless  telegraphy  she  signals  a  tug, 
which  comes  out  to  meet  her  at  17  knots  an  hour.  In  how 
long  a  time  will  they  meet  ?  If  the  steamer  is  s  knots  from 
port  and  is  making  Vx  knots  per  hour,  and  if  the  tug  makes  Vz 
knots  per  hour,  find  how  long  before  they  will  meet. 

27.  A  motor  boat,  going  11  miles  per  hour,  starts  7|-  miles 
behind  a  sailboat  going  6i  miles  per  hour.  How  far  apart  will 
they  be  in  1^  hours  ?  If  the  motor  boat  starts  s  miles  behind 
the  sailboat  and  runs  v^^  miles  per  hour,  while  the  sailboat  runs 
Vo  miles  per  hour,  how  far  apart  will  they  be  in  t  hours  ? 

28.  An  ocean  liner  making  21  knots  an  hour  leaves  port  when 
a  freight  boat  making  8  knots  an  hour  is  already  1240  knots  out. 
In  how  long  a  time  will  the  two  boats  be  280  knots  apart  ?  Is 
there  more  than  one  such  position  ?  If  the  liner  makes  Vi  knots 
per  hour  and  the  freight  boat,  which  is  s^  knots  out,  makes  V2 
knots  per  hour,  how  long  before  they  will  be  So  knots  apart  ? 

29.  A  passenger  train  running  45  miles  per  hour  leaves  one 
terminal  of  a  railroad  at  the  same  time  that  a  freight  running 
18  miles  per  hour  leaves  the  other.  If  the  distance  is  500  miles, 
in  how  many  hours  will  they  meet  ?  If  they  meet  in  8  hours, 
how  long  is  the  road  ?  If  the  rates  of  the  trains  are  Vi  and  V2 
and  the  road  is  s  miles  long,  find  the  time. 

30.  The  melting  temperature  of  glass  is  276  degrees  (Centi- 
grade) higher  than  twice  that  of  zinc.  One  half  the  number  of 
degrees  at  which  glass  melts  plus  7  times  the  number  at  which 
zinc  melts  equals  3434.     Find  the  melting  point  of  each. 

31.  The  melting  temperature  of  nickel  is  496  degrees  (Centi- 
grade) higher  than  that  of  silver.  Three  times  the  number  of 
degrees  at  which  nickel  melts  plus  2  times  the  number  at  which 
silver  melts  equals  6258.     Find  the  melting  point  of  each. 


COLLEGE   ENTRANCE   EXAMINATIONS  353 

HARVARD   UNIVERSITY 


ELEMENTAEY  ALGEBRA 


TniE  :  One  Houk  and  a  PLvlf 

1.  Solve  the  simultaneous  equations 

,   ,  y  -\-h      a 

X  -{-  a      0 
and  verify  your  results. 

2.  Solve  the  equation  x"^  —  1.6  ic  —  0.23  =  0,  obtaining  the 
values  of  the  roots  correct  to  three  significant  figures. 

3.  Write  out  the  first  four  terms  of  (a  —  hf. 
Find  the  fourth  term  of  this  expansion  when 

3/ 1  Q     

a='^  x-^y^-,  b=w9xy-\ 

expressing  the  result  in  terms  of  a  single  radical,  and  without 
fractional  or  negative  exponents. 

4.  Reduce  the  following  expression  to  a  polynomial  in  a 

and  b : 

6  a^  +  T  ab'^  +  12  b^ 1 

3a2-5a6-4&2         3        5  a  +  4  6 ' 
19  6         19  a2 

5.  The  cost  of  publishing  a  book  consists  of  two  main  items  : 
first,  the  fixed  expense  of  setting  up  the  type  ;  and,  second,  the 
running  expenses  of  press  work,  binding,  etc.,  which  may  be 
assumed  to  be  proportional  to  the  number  of  copies.  A  certain 
book  costs  35  cents  a  copy  if  1000  copies  are  published  at  one 
time,  but  only  19  cents  a  copy  if  5000  copies  are  published  at 
one  time.  Find  (a)  the  cost  of  setting  up  the  type  for  the 
book,  and  (b)  the  cost  of  press  work,  binding,  etc.,  per  thou- 
sand copies. 


354  COLLEGE   ENTRANCE   EXAMINATIONS 

YALE   UNIVERSITY 


ALGEBRA   A 


Time  :  One  Hour 

Omit  one  question  in  Group  II  and  one  in  Group  III.     Credit  will  be 
given  for  six  questions  only. 

Group  I 

1,  Kesolve  into  prime  factors  :  [a]  6  a;^  —  7  x  —  20 ; 
ip)  (x2-5a;)2-2(aj2_5x)-24;  (c)  a^  +  4  a^  +  16. 

2.  Simplify  (^  -  ^'-^^^'\ ^fs      «  "  '^  ^ 


Ax^  J     \        a  —  2x 

3.  Solve  2(0.-7)      ^2-x_xJ^^^^ 

x2-\-3x-2S      4.-X      x-}-7 

Group  II 

a/2  -I-  2  V3 

4.  Simplify  ■ — ^^ — =^,  and  compute  the  value  of  the  frac- 

V2  -  Vl2 
tion  to  two  decimal  places.  .    _i  _^ 

5.  Solve  the  simultaneous  equations      '  if  ^—  'si 

(2a;-^-2/-^  =  |. 

Grouj)  III 

6.  Two  numbers  are  in  the  ratio  of  c  :  d.  If  a  be  added  to 
the  first  and  subtracted  from  the  second,  the  results  will  be  in 
the  ratio  3  :  2.     Find  the  numbers. 

7.  A  dealer  has  two  kinds  of  coffee,  worth  30  and  40  cents 
per  ])ound.  How  many  })()un(ls  of  each  must  be  taken  to  nuike 
a  mixture  of  70  pounds,  worth  36  cents  per  pound  ? 

8.  A,.  B,  and  C  can  do  a  piece  of  work  in  30  hours.  A  can 
do  half  as  much  a^ain  as  B,  and  B  two  thirds  as  much  again 
as  C.     How  long  would  each  require  to  do  the  work  alone  ? 


COLLEGE   ENTRANCE   EXAMINATIONS  355 

CORNELL  UNIVERSITY 


ELEMENTARY   ALGEBRA 


1.  Eind  the  H.  C.  E.  : 

x^  —  xy-  -{-  x-y  —  y^, 

2.  Solve  the  following  set  of  equations : 

x  +  y  =  -l, 

X  —  y  -\-  4:Z  =  D. 

3.  Expand  and  simplify  : 


2x^-^' 

4.  An  automobile  goes  80  miles  and  back  in  9  hours.  The 
rate  of  speed  returning  was  4  miles  per  hour  faster  than  the 
rate  going.     Find  the  rate  each  way. 

5.  Simplify:         /x  +  IV      o   ,  A'  -  1 V 

X 


x-\-iy-    fx  -  iv 

x-lj       \x  +  lj 


6.    Solve  for  x : 


2a;4-3      g^  5 

x-1  x'^-\-2x-3 


7.  A,  B,  and  C,  all  Avorking  together,  can  do  a  piece  of 
work  in  2f  days.  A  works  twice  as  fast  as  C,  and  A  and  C 
together  could  do  the  work  in  4  days.  How  long  would  it 
take  each  one  of  the  three  to  do  the  work  alone  ? 


356  COLLEGE   ENTRANCE   EXAMINATIONS 

UNIVERSITY  OF  CHICAGO 


ELEMENTARY   ALGEBRA 


I.   Resolve  into  simplest  factors  : 
1.    a'-b^-ia-by. 

3.  x^  —  14  xY  +  y^' 

4.  a{a  +  6)  -  c{c  +  b). 

11.    Reduce  to  simplest  form  : 

1.  ^io\x-6[x-(ij-z)^\  ^60\y-(z-[-x)\. 

2  ^^  ,  ^.  I ^ 

(6  -  c){b  -a)      (c-  a)(c  -  b)      (b  -  a)(c  -  a) 

3  ^«+LVA  _1  .  L 

[b^     ay  '  Va'      ab      b^ 

III.  1.    Solve  for  x: 
^(x  -  5)  -  i(x  -  4)  =  ^^{x  -  3)  -  (X  -  2). 

2.  Solve  for  x  and  2/ : 
i(7  +  x)  =  4(9  +  ?/). 

IV.  Three  brothers,  A,  B,  C,  at  a  family  reunion  were  dis- 
cussing their  ages.  C  said  to  A :  "  Thirty  years  ago  my  age 
was  double  yours."  Then  B  said  to  A  :  "  Twenty-three  years 
ago  my  age  was  double  yours."  If  C's  present  age  exceeds  B's 
by  three  years  and  B's  exceeds  A's  by  seven  years,  find  the 
age  of  each. 


COLLEGE   ENTRANCE   EXAMINATIONS  357 

UNIVERSITY  OF  CALIFORNIA 


ELEMENTARY   ALGEBRA 


1.  If  a  =  4,  6  =  —  3,  c  =  2,  and  d  =  —  4,  find  tlie  value  ot : 

(a)  ab'  -  3  ccP  +  2(3  a-b)(c-2  d). 

(b)  2  a'  -Sb'-h  (4  c^  +  (J^)  (4  c"  +  r?^). 

2.  Reduce  to  a  mixed  number : 

3  a^  _  4  a^  -  10  a2  +  41  a  -  28 


a2_3a  +  4 

Simplify : 

a  +  2 

b-2 

a2  +  3  a  -  40      ab  -  o  b -{- S  a  -  15 
^_2  -Sb-2c\  .  a2  -  4  c2  +  9  &2  +  6  «6 


(1+2       y  2a2  +  a-6 

5.  A's  age  10  years  hence  will  be  4  times  what  B's  age  was 
11  years  ago,  and  the  amount  that  A's  age  exceeds  B's  age  is 
one  third  of  the  sum  of  their  ages  8  years  ago.  Find  their 
present  ages. 

6.  Draw  the  lines  represented  by  the  equations 

ox  — 2  y  =  13  and  2  a;  +  5  ?/  =  —  4, 
and  find  by  algebra  the  coordinates  of  the  point  where  they 
intersect. 


^  ,  •    ^,  ^.        f  bx  —  ay  =  6'  —  ab, 

7.  Solve  the  equations  V      .^/        on 

^  [      y  —  b  =  2{x  —  2  a). 

8.  Solve  (2  a;  + 1)(3  a;  -  2) -  (5  x  -7){x-2)  =  41. 


358  COLLEGE   ENTRANCE    EXAMINATIONS 

NEW    YORK    STATE    EDUCATION    DEPARTMENT 


ELEMENTARY   ALGEBRA 


January,  1912 
Ansioer  the  first  six  questions  and  two  of  the  others. 

1.  Find  the  prime  factors  of  each  of  the  following:  1  —  x*; 
x^-cx-2dx  +  2cd;  3  a^  +  3  6^ ;  a^  +  a'h''  +  h\ 

2.  Divide  4  a^  -  9  a;^  +  25  -  14  a^  -  a^^  ^y  2  a^  -  a;  -  5+3  x\ 

3.  Solve   \^nx-ny  =  m^  +  n\ 

[       X  —  y  =  2  n. 

4.  Simplify  V3  x  V48  ;  (5  V5  -  4)(5  V5  +  8) ; 
3  V8  - 15  V2  ;  6  V|  -  5  V24  +  12  Vf. 

5.  Find  the  square  root  of  the  following : 

25  x^  -  30  aa^  +  49  aV  -  24  a^x  +  IG  a\ 

6.  The  area  of  a  rectangle  is  18  square  inches  less  than 
twice  the  area  of  a  square ;  the  width  of  the  rectangle  equals 
the  width  of  the  square,  but  the  length  of  the  rectangle  ex- 
ceeds that  of  the  square  by  7  inches.  Find  the  side  of  the 
square. 

7.  A,  B,  and  C  together  have  $1285;  A's  share  is  $25 
more  than  f  of  B's,  and  C's  share  is  y\  of  B's.  Find  the  share 
of  each. 


8.  Solve  21  a;2  =  2ax  +  ?>  a". 

9.  Solve    \^-^y+f  =  ^, 

[  X  —  //  =  —  :>. 


10.    Define    root    of    a    number,    surd,    affected    quadratic 
equation. 

Expand  {2ar-\^by. 


COLLEGE    ENTRANCE   EXAMINATIONS  359 

NEW   YORK    STATE    EDUCATION   DEPARTMENT 


ELEMENTARY   ALGEBRA 


June,  1912 
Ayiswer  the  first  four  questions  and  four  of  the  others. 

1.  Divide  9  x^ -lOa:^ -\-9  -  16  x"^  -  x^  hj4:X-x'^-{-Sx^- 
3.  [Credit  will  not  be  granted  if  there  is  any  error  in  the 
work.] 

2.  Factor  each  of  the  following  :  x*  —  x^  —  12  ; 

(a  +  2)2  -  9  ic2 ;  cf^  -  27  ;  6x'-x-2;  a;^  +  32  ;  a' -  a^  -  a+l. 

3.  Solve  -^  ' 

[  X  —  2  y  =  a  —  1. 

4.  The  length  of  a  rectangular  field  is  twice  its  width ;  it 
costs  as  much  to  fence  it  at  50  ^  per  yard  as  it  does  to  sod  it 
at  15  ^  a  square  yard.     Eind  the  dimensions. 


5.  Solve2+V2a;-f8  =  2V:«  +  5. 

6.  Solve  x^  +  ax  =  42  a^. 

Exj)and  by  the  binomial  formula  (2  a  —  3  6)^,  giving  all  the 
work. 

7.  Solve  [^'  +  f  =  ^^^ 

[xij  =  6. 

Group  the  four  pairs  of  roots  properly. 

8.  Two  men  start  at  the  same  time  and  travel  in  opposite 
directions  ;  the  ratio  of  their  rates  is  2  :  3  and  in  5  hours  they 
are  100  miles  apart.     Find  the  rate  of  each. 

9.  Divide  the  number  c  into  two  parts  such  that  a  times 
the  larger  part  shall  e([ual  h  times  the  smaller  part. 

10.  If  a  triangle  with  equal  sides  has  its  sides  increased  7 
inches,  4  inches  and  1  inch  respectively,  a  right  triangle  is 
formed  ;  find  the  sides  of  the  right  triangle. 


360  COLLEGE   ENTRANCE   EXAMINATIONS 

NEW    YORK   STATE    EDUCATION    DEPARTMENT 


ELEMENTARY   ALGEBRA 


January,  1913 
Ansioer  the  first  six  questions  and  two  of  the  others. 
1.   Factor  each  numerator  and  denominator  in  the  following 
expressions ;  perform  the  operations  indicated  and  reduce  to 
simplest  form : 

a^  _  9  a^  _  36  a.'2  x^-^x^A-3    .      o^  +  x- 


x^a^  -  10  xa"^  +  9  a2     a?^  -  7  .t^  -  18      aV  +  2  a^ 
2.    Solve 


o  "^  5  —  ^?  3.    Solve  a-  +  -  = 

2     3  ^2      2x 

5x-32j=2. 


Simplify  each  of  the  following : 


4V24  +  2V54  -  V6  +  3  V96  -  5  Vl50. 

3, /2      '/I      1 

5.  Solve  i^i^+?-^:^^  =  ^^iL^. 

10  5aj-  1  5 

6.  Separate  42  into  tico  parts  such  that  the  greater  part 
divided  by  the  less  shall  give  a  quotient  of  2  and  a  remainder 
of  3. 


7.    Solve    f62/^-a:y  =  2a.^ 
l9y-12  =  -4a; 


9y/-12  =  -4a;. 

8.  What  must  be  added  to  ic  +  a  to  make  ?/  —  b? 
What  is  the  cost  of  3  apples  if  a  apples  cost  c  cents  ? 

9.  The  sum  of   two  numbers  is  8  and    the  sum  of    their 


cubes  is  152  ;  find  the  numbers. 


10.  If  1  is  added  to  the  numerator  of  a  fraction  the  value  of 
the  fraction  becomes  i;  if  1  is  added  to  the  denominator  of 
the  same  fraction  the  vahie  becomes  ^.     Find  the  fraction. 


COLLEGE   ENTRANCE   EXAMINATIONS  361 

NEW   YORK    STATE    EDUCATION    DEPARTMENT 


ELEMENTARY   ALGEBRA 


Junk,   1913 
Answer  the  first  six  questions  and  tioo  of  the  others. 

1.  Solve  and  check  or  prove  -  —  ^  ~     =  12  —  ^^ x. 

^33  2 

2.  Extract  the  square  root  of  4  c*  —  Ic^+Sc^  —  2c-i-l. 

^  ,        f  a;  4-  2  7/  =  a, 

3.  Solve    '     ^     ^'        ' 


2x-y  =  b. 

4.  Simplify  Vf  +  V|;  (V5  -  V2)(2V5  +  3V2). 

5.  In  five  years  A  will  be  twice  as  old  as  B  ;  five  years  ago 
A  was  three  times  as  old  as  B.  Find  the  age  of  each  at  the 
present  time. 

6.  Find  the  quotient  to  thi^ee  terms  and  the  remainder  when 
11  ci^  _  5  ct-h  12  -  82  a2  +  30  a^  is  divided  by  2  a  -  4  +  3  a\ 

7.  (a)  What  is  the  dividend  which,  divided  by  x,  gives  a 
quotient  of  y  and  a  remainder  of  z  ? 

(b)  If  a  apples  are  sold  for  a  dime,  how  many  can  be  bought 
for  c  cents  ? 

8.  Two  men,  A  and  B,  can  dig  a  trench  in  20  days  ;  it 
would  take  A  9  days  longer  to  dig  it  alone  than  it  would  B. 
How  long  would  it  take  B  alone  ? 

9.  Find  three  successive  even  numbers  whose  sum  is  |  of 
the  product  of  the  first  two. 

2      1 

x^ 

X 

10.    Simplify z • 

X 


362  COLLEGE   ENTRANCE    EXAMINATIONS 

THE    UNIVERSITY    OF   THE    STATE    OF    NEW    YORK 


ELEMENTARY   ALGEBEA 


January,  1910 
Ansiver  eight  questions,  selecting  tico  from  each  group. 


1.    Solve 


Group 

a  -\-hx  a—  hx 


3  6  +  2  ax     b  —  2  ax 

2.  Find  the  prime  factors  of  each  of  the  following  expres- 
sions and  from  the  factors  determine  the  highest  common  factor : 

27  m^  —  8  m^,  6  m^  +  8  m"^  —  8  m,  12  m*  —  8  m^,  27  m^  —  12  m. 

3.  Find  the  product  of  2  x  -^3  y—  z  and  x  —  3  y  -\-  2  z. 
Prove  by  substitution  the  correctness  of  the  result  if  x  =  l, 
y  =  2,  and  z  =  3. 

Group  II 

4.  Eeduce  each  of  the  following  to  its  simplest  form : 

V50  -  V32,  2V5  X  Vl5,  6V20  -  2V10,  V|,  VoVa^ 
f        2  a;  +?/  —  2:  =  5, 

5.  Solve     3x+2y  —  ^z  =  2, 

x-2y  +  3z  =  3. 

6.  Find  the  square  root  of  the  following : 

49  x^  -  42  a;^  -  47  a;4  -  4  o9  +  28  x"-  +  16  x  +  4. 

Group  III 

7.  If  122  marbles  were  divided  among  three  boys  so  that 
the  first  had  twice  as  many  as  the  second  and  the  second  had 
6  more  than  the  third,  how  many  had  each  ?     Prove. 

8.  Find  three  consecutive  numbers  sucli  that  if  the  lirst  is 
divided  by  5,  the  second  by  7,  and  the  largest  by  11,  the  sum 
of  the  three  quotients  is  \  of  the  sum  of  the  three  numbers. 
Prove. 


COLLEGE  ENTRANCE  EXAMINATIONS       363 


9.    Solve  V2ic  +  l  =  2Va;-Vaj-3. 

Group  IV 

10.  Solve  j2^  +  ?V  =  5' 

11.  Solve  1  -  10  a.^  +  16  aV  =  0. 

12.  A  and  B  can  together  address  100  envelopes  in  an  hour  ; 
when  each  works  alone  A  can  address  100  envelopes  in  50 
minutes  less  time  than  B.  How  many  can  each  address  in 
an  hour  ? 

THE    UNIVERSITY   OF    THE    STATE    OF    NEW    YORK 


ELEMENTARY   ALGEBRA 


January,   1915 

Ansioer  12  questions^  selecting  five  from  group  /,  two  from  group  11^ 
and  five  from  group  III. 

Group  I 

1.  Solve  ^(^-^)  =  ^^-^-^-^-+li^\ 

x  +  3  a;2-9         3-a; 

2.  Factor  three  of  the  following : 

x^  -  8  x"  -  9, 

m^  —  6  mn  —  16  x-y"^  +  9  n^, 
m^d^  +  3  -  3  771  -  cV, 
p'^q^  —  12  pqx  +  35  Qi?. 

3.  Solve  V3T^-  +  Va-  =  -4=  • 

■\'x 

4.  Solve3(.T-2)(.i--4)  =  {.f-5)2. 

5.  (a.)    Simplify  2 V|-V60- 5 Vf. 
(6)    Simplify  «V^-^^^. 


364       COLLEGE  ENTRANCE  EXAMINATIONS 

6.  Solve  |^-3.V  =  1, 

I  xy  -h  ?/2  =  5. 

Group  II 

7.  A  man  has  $8000  which  he  wishes  to  invest  in  two 
enterprises  so  that  his  total  income  will  be  $  425  ;  if  one  en- 
terprise pays  5.^  %  and  the  other  5  % ,  how  much  must  he  in- 
vest in  each  ? 

8.  Separate  a  line  20  inches  long  into  two  parts  such  that 
the  product  of  the  whole  line  and  one  part  shall  equal  the 
square  of  the  other  part.     [Result  contains  a  surd.] 

9.  A  rectangular  ceiling  has  in  it  two  skylights  each  2i  feet 
by  3  feet ;  the  surface  of  the  ceiling,  not  including  the  sky- 
lights, is  93  feet.  If  the  length  of  the  ceiling  is  3  feet  more 
than  its  width,  what  are  its  dimensions  ? 

Group  III 

10.  If  one  of  the  factors  of  6  a^x-  —  4  o?x  —  4  ay?  +  if*  +  a^  is 
a2  _|_  .^2  _  2  ax,  what  is  the  other  factor  ? 

11.  Write  three  different  expressions  of  higher  degree  than 
the  first  degree  whose  H.  C.  F.  is  x—  y.  Find  the  L.  C.  M.  of 
these  expressions. 

12.  A  lady  bought  »5  dozen  buttons  at  d  cents  a  dozen  and  3 
yards  of  cloth  at  A'  cents  a  yard  ;  she  gave  a  two-dollar  bill  in 
payment.     How  many  cents  should  she  receive  in  change  ? 

13.  What  is  the  value  of  8  x^  —  6  ax  when  x  =  - — ~ —  ? 

14.  Find,  correct  to  two  decimal  ])l;i(*es,  the  solutions  of 
2  x"  +  6  ■^•  -  3  ==  0. 

15.  Write  an  expression  that  cxui  be  divided  by  a  —  h  and 
also  by  2  a  +  6. 


COLLEGE   ENTRANCE   EXAMINATIONS  365 

THE    UNIVERSITY    OF   THE    STATE    OF   NEW    YORK 


ELEMENTARY   ALGEBRA 


June,  1915 
Ansioer  the  first  eight  questions  and  two  of  the  others. 

1.  Find  tlie  prime  factors  of  each  of  the  following  : 

27  -  64  0.-3 ;  10  a;2  -  7  a;  -  6  ;  ax^  —  ex  +  ax  —  c. 

2.  When  a  =  2,  6=3,  c=  —  4,  find  the  value  of  the  following : 

(3  a2  _  c2)(a  -f  c)V(7a  +  c)(4  6-a). 

3.  Solve   a;2  —  4  oj  —  1  =  0  ;   find  the  roots  correct   to   two 
places  of  decimals. 

4.  Solve  Vic  +  IG  —  V.^  =  2. 

5.  (a)  Simplify  V48  -  2V45  +  lOVX- V|. 
(6)   Simplify  (3 V5  -  2 V2)(2V5  +  4 V2). 

a2-f-62       7.    Solve  L    ,  .  ,o     .  .        q 
y  —  x  =  — -^ 3a;2-f  12a;+4?/=8. 


6.    Solve 


8.  A  man  has  oats  enough  for  x  horses  for  y  days ;  how 
long  will  the  oats  last'^^i  horses  ? 

9.  If  the  greater  of  two  numbers  is  divided  by  the  less, 
the  quotient  is  2  and  the  remainder  is  1.  If  the  less  is  in- 
creased by  20  and  this  result  is  divided  by  the  greater  increased 
by  3,  the  quotient  is  2.     Find  the  numbers. 

10.  What  must  be  the  value  of  m  in  order  that 

2  ir2  _  3  ^^  _^  21  a;  +  .c*  -f  3  m 
may  be  exactly  divisible  by  x^  —  S  -\-  x'.* 

11.  (a)  What  value  of  x  will  make  (6  x  —  5)  (2  x  —  3)  equal 
to  13  more  than  (3  a;  +  2)  (4  x  -  1)  ? 

(6)    How  many  dimes  taken  from  a  dollars  will  leave  10  a; 
cents? 


366  COLLEGE   ENTRANCE   EXAMINATIONS 

THE    UNIVERSITY   OF   THE    STATE    OF    NEW   YORK 


ELEMENTARY   ALGEBRA 


January,  1914 
Answer  the  first  six  questions  and  two  of  the  others. 

1.   Find  the  prime  factors  of  a;^—  x  —  30 ;  16  —  2  m^ ;  1  —  ot^-, 
2  ab  —  ex  -\-  2c  —  abx. 


2.  Subtract  —bx  —  2{&y—2z)  from  ^x  —  z-\-2y  and  add 
the  result  to  Q>  z  —{3y  —  ^x). 

3.  Simplify  f-?-Y-^^  +  -^^±^ ""-y   \ 

V^  +  yA^'  -y'    K^  -  y)    ^^  +  y)J 
«-  —  =  -• 

4.  Solve  for  a  and  b  2       2 

3a  4-7  5  =  13. 

5.  Solve  (x  -  6)2  -{2x-  5y~  =  16. 

6.  One  number  is  twice  another  number  ;  when  the  smaller 
is  subtracted  from  32  the  remainder  is  11  less  than  the  re- 
mainder when  the  larger  is  subtracted  from  50.  Find  the 
numbers. 

7.  Solve  and  test 

3(x  +  l){x  -  3) -  (3  -  xy  =  8(2  X  -3)-{-2x'~-  26. 

8.  (a)  Reduce  to  radicals  of  the  same  degree  V2,  V3,  v  2. 
(b)  Perform  the  indicated  operations  (2  —  V5)2(l  —  2 V5)  ; 

2V6--^3. 

9.  The  length  of  a  rectangle  is  15  ft.  greater  than  its  width  ; 
if  each  dimension  is  decreased  2  ft.,  the  area  will  be  decreased 
106  sq.  ft.     Find  the  dimensions. 

10.  (a)  The  dividend  is  m,  the  divisor  is  n,  the  remainder  is 
r ;  what  is  the  quotient  ? 

(6)  A  house  cost  a  dollars  and  rents  for  7i  dollars  a  month ; 
what  per  cent  per  annum  is  the  income  of  the  investment  ? 


sions 


COLLEGE   ENTRANCE   EXAMINATIONS  367 

THE   UNIVERSITY   OF  CHICAGO 


ELEMENTARY   ALGEBRA 


I.    Remove  the  symbols  of  aggregation  from 

1.  2x-]3y-[oz-(x-2y-3z)']\', 

2.  2  x  +  \5  z  -[3  y  -\-(x  -  2  y  -~  3  z)]\. 


II.    Arrange  the  following  in  groups  of  equivalent  expres- 


X     X  -\-  a     xa     X  —  a     x  -i-  a^ 

a  X     y      y     a 

a  XX  y     o. 


III.   Eind  the  G.  C.  D.  of  : 

2  ax^  -f  2  ax"^  —  4  ax, 

(aa;2  +  2ax)(2ar»_2), 

2a{x^-x){x?-\-^), 

(x  —  1)(6  abx^  + 12  abx  —  2  ax'^y  —  4  axy). 


lY.   Solve  for  x : 

2.T-1  23  3/    1         1 


2ic-2      10a;-10      5Va;-l      3 


Y.  Two  yachts  race  over  a  48-mile  course.  Owing  to  dif- 
ference in  measurement,  B  is  given  a  start  of  half  a  mile  in 
the  first  trial,  and  is  beaten  by  6  minutes.  In  the  second 
trial,  the  rate  of  the  wind  being  the  same  as  before,  B's  start 
is  increased  to  a  mile  and  a  half,  and  still  A  wins  by  2  minutes. 
Eind  the  rate  in  feet  per  minute  of  each  boat. 


INDEX 


[References  are  to  pages.] 


Abscissa 233  j  Cajori 


62 


Absolute  value 44 

Addition 8 

by  counting 45 

of  fractions 168 

of  imaginaries 284 

of  polynomials 69,  71 

of  signed  numbers      ...     45,  47 

of  surds 268 

principles  of 9,  47 

Ahmes 41 

Algebra  (origin  of  name) ...      32 
Algebraic,  fractions     ....    160 

operations 20 

sum 48 

Algebraic  expressions  ...        6 
forming  of,  7,  37,   79,    81,  87, 

91,   92,  161,  220,  242,  291 

Alkarismi 101 

Alternation 189 

Antecedent 188 

Approximate  roots    .     .     .  251, 253 

Arabs 5,32,41,62,313 

Arabic  numerals 5 

Area  problems,  1,  2,  3,  4,  87, 
91,  120,  154,  220,  221,  222, 
223,  259,  260,  261,  294,  295, 
305,  306,  323,  337,  :^7,  ;348. 

;349.  350,  351 
Averages  of  signed  numbers 

49,  (>4 
Axes  of  coordinates  .     .      232,  233 


Base  of  a  power      .     , 

Bhaskara 

Binomial 

Binomial  formula  .     , 
Braces,  brackets,  etc. 


.  92 
.    291 

85,  108 
.  308 
.      11 


Checking  results,  10,  27,  32, 

33,  34,69,  102,211,281,  282 
Circle,  problems  on,   261,    317, 

339,  347,  348,  349,  350 
Clearing  of  fractions      .     .     .    182 

Coefficient 8,  92 

Common,  denominator     .     .     .    164 

factor 8,  155 

multiple 157 

Commutative  laws     .     .     .     69. 84 
Completing  the  square      .     .    278 

Hindu  method  of 280 

Complex  fractions     ....    178 

Consequent 188 

Contradictory  equations    207,  238 

Coordinates 233 

Cross  products 134 

Cube  of  binomial    .     .   116,  308,  309 
Cubes,  sum  or  ditference  of  .     .     118 

Data    problems,   39,  227,   228, 

319,  322,  324,  327,  S44,  351,  352 
Degree  of  an  equation,   147, 

207,  236,  289 
Deriving  eqmvalent  equa- 
tions .  .  .27,  28,  29,  256,  276 
Dependent  equations  .  207,  2.i8 
Descartes  .  .  41,  62, 101,  239,  2i>l 
Detached  coefficients    .     .    .    106 

Difference 9,  50 

of  two  cubes 118,  129 

of  two  squares  ....       110,  126 
Digit  problems,  81,  82,  83,  222, 

223,  228,  343,  351 
Diophantus  ...  41,  62,  229,  291 
Directions  for  written  work  30 
Distributive  laws 15 


369 


370 


INDEX 


[References  are  to  pages.] 


Dividend 59 

Division 14,  18,  59 

by  a  monomial 96,  98 

by  a  polynomial 102 

by  zero 29,  102,  104 

indicated 14,  21 

of  fractions 175 

of  imaginaries 285 

of  a  product 18 

of  signed  numbers      ....      59 
of  sum  or  difference  ....      14 

of  surds 274 

principles  of  .     .     .     .15,  18,  59,  97 

Divisor 59 

Double  use  of  signs    ...     52,  53 


Egyptians 41,  151, 

Entire  surds 

Equations 

contradictory     ....      207, 

degree  of  .     .     .     147,  207,  236, 

dependent 207, 

equivalent 

graphic  representation  of   .     . 

graphic  solution  of     ...     . 

homogeneous 

identical 

independent   ....   208, 

indeterminate 

involving  fractions     .... 

involving  radicals      .... 

linear 

literal,  1,2,  3,4,  188,  190,  191, 
196,  198,  199,  201,  203,  204, 
205,  206,  217,  218,  219,  229, 

quadratic 147,  278, 

roots  of 

simultaneous 

solution  of,  m,  27,  28,  29,  54, 
147,  181,  207,  208,  211,  224, 
237,  256,  278, 

solved  by  factoring     .  147,  148, 

translation  of 

Exponents  

fractional 


161 

262 
11 
238 
289 
238 
25 
236 
237 
301 
11 
238 
207 
181 
276 
236 


283 

297 

26 

207 


289 

297 

32 

92 

26(> 


negative 100 

zero 100 

Expressions,  algebraic  ...        6 

Factoring 122 

cases  of 142,  143 

Factors 8,  122 

highest  common 155 

Falling  body  problems .     .    .    293 

Form  changes 27,  29 

Formulas,  1,  2,  4,  121,  196,  198, 
199,  201,  203,  204,  206,  217, 

259,  261,  283,  308,  311 
Fourth  proportional  ....  190 
Fractional  equations      .     .     .     181 

Fractions 160 

clearing  of 182 

complex 178 

reduction  of 162,  164 

signs  of 166 

Functions 314 

decreasing 315 

increasing 314 

Fundamental  laws .     .     .     .    15, 69 

Girard 11 

Graphic    representation,    of 

e(iuati()us 235 

of  signed  numbers     '.     .     .     .      44 

of  .statistics 230 

Greeks 62,  161,  206,  229 

Hamilton 70,  84 

Harriot    .     .     .62.  101,  206,  239,  291 
Highest  common  factor    .     .    155 
Hindus     .     .    r..  62,  161,  291,  280,  313 
Historical  notes,  5, 7,  11, 15,  32, 
41,  62,  69,  SI,  101,  151,  161, 

188,  206,  239,  242,  291,  313 
Homogeneous  equations  .     .    301 

Identities 11 

Imaginary,  number      ....    284 
roots  of  a  quadratic   ....     287 

Index 92,262 

Inserting  in  parentheses    .    .      77 


INDEX 


371 


[References  are  to  pages.] 


Integrera,  even,  odd 38 

problems  on,  'M,  4<J,  'Jl,   120, 

2i>i,  :ny,  320,  32« 

Interest  problems,  4,   '.Mi,  41, 

I'.tT,  •_"_'(».  221,  ;51(>,  339 
Inversion  of  a  proportion  .     .     IH\) 

La-ws  of  exponents,  'M,  '.»."),  \)7, 

2«kj,  2G7 

Leonardo  of  Pisa 5 

Lever  problems     .     .    202,  224,  345 
Linear  and  quadratic  equa- 
tions     21t7 

Linear  equations 23«i 

Literal  equations,  1,  2,  3,  4,  181), 
1!K),  191,  196,  198,  im,  201, 
203,  204,  205,  206,  217,  218, 

219,  229,  283 

Long- division 102 

Lo^west  common  multiple  157 

Lowest  terms,  fractions  in    .     1()2 

Mathematical  induction  .  .  310 
Mean  proportional  ....  190 
Means  of  a  proportion  .  .  .  ISS 
Members  of  an  equation    .     .      11 

Minuend 50 

Monomial 67 

Motion  problems,  199,  200,  201, 

202,  222,  223,  224,  332,  343, 

a44,  345,  349,  350,  351 

Multiple 

lowest  coiiimon       .     .     .       157 
Multiplication 1 


of  fractious    .     . 
of  imagiuaries   . 
of  polynomials  . 
of  signed  number.- 
of  surds     .     .     . 


,  352 

157 

,  158 

2,  16 

172 


285 
84 
56 

270 


Negative,  exponents    ....     100 

number 43,  44,  62 

results,  interpretation  of    .     .       63 

Number,  absolute  value  of  .  .  44 
positive  and  negative      .     .     43,  44 


rational 262 

unknown 25 

Number   system. 


of  .\rithnu'tic 
NeTvton     .    . 


of    alj^ebra 

44,  62,  2M 

44 

62,  101,  20<;,  313 


Operations,  at  sij^ht     ....  31 

ortler  of 20 

sytnliols  ()f ;;.  7 

Ordinates 2:33 

Oughtred 7 


Parentheses 11 

removal  of 

insertion  in 

Pascal 

Percentage  problems    .     .      4, 

Polynomials 


adtlition  and  subtraction  of    69 

division  by 

multiplication  of    ....     85, 

square  root  of 

Positive  numbers   ....     43 

Power .     . 

Prime  factor 

Principles,  eii^hteen  fundamen- 
tal.   See  List,  page  ix. 

importance  of 

Problems 

involving  data,   39,  227,   228, 

319,  322,  324,  327,  351, 
involving     the     Pythagorean 
proposition,   152,   153,   257, 
258,  259,  260,  292,  293,  30«>. 
307,  Ml, 
on  areas,  1,  2,  3,  4,  87,  91,  120, 
1.54,  220,  221,  222,  223.  259. 
2»iO,  261.  21  >4,  295,  305,  306, 
323.  .3.37.  :«7,  348,  ^49,  liTyO, 
on  circles,  261,  317,  339.  ^7, 

MS.  349, 
on  digits,  SI,  82,  83,  222.  223. 

228,  MS, 
on  falling  bodies 


313 
:i50 

67 
.  72 
102 
,88 
243 
,  44 

92 
122 


352 

342 

351 

350 

351 
293 


372 


INDEX 


Problems  {Contimied) 

on  integers,  38,  40,  91, 120,  291, 

319,  320,  326 
on  interest,  4,  36,  41,  197,  220, 

221,  316,  339 

on  levers 203,224,345 

on  motion,  199,  200,  201,  202, 

221,  222,  223,  234,  332,  343, 

:U4,  345,  349,  350,  351,  352 
on  percentage    .     ....     4,  350 

on  ratios        192 

on  rectangles,  1,  3,  4,  38,  39, 
40,  87,91,120,  154,  220,221, 

222,  223,  292,  294,  295,  305, 
306,  319,  337,  338,  341,  347,  &48 

on  similar  triangles       .     .     .      193 
on  triangles,  2,  3,  92,  193,  194, 
195,  257,  258,  259,  260,  261, 
293,  294,  306,  307,  341,  342,  347 
on  volumes,  2,  3,  4,  36,  153, 

295,  325 

Products,  special 108 

Proportion 188 

Pythagorean  theorem  .     .     .     151 
problems  on,  152,  153,  257,  258, 
259,  260,  292,  293,  306,  307, 

341,  342 

Quadrants 233 

Quadratic  equations      .      147,  278 
equations  in  the  form  of      .     .     289 

homogeneous 301 

solution    by    completing    the 

square 279,  280 

solution  by  factoring      .     .     .     147 
solution  by  formula   .     .      283,  299 

systems  of 297 

Quadratic  surds     c     •    262,  271,  274 
Quotients,  special 117 

Radical  expressions  ....    2(52 

Radicals  .......      240,  262 

ai)plications  of  .     .     .     .     .     .    257 

equations  involving    .     .      276,  2i)0 
simplification  of    .    c    .     .    .    252 


[References  are  to  pagres.] 

Radicand 262 

Rahn 7 

Ratio 188,  192 

Rational,  numbers 262 

roots 262 

Rationalizing"   denominators 

254,  274 

Real  numbers 284 

Rectangles,  problems  on,  1,  3, 
4,  38,  39,  40,  87,  91,  120,  154, 
220,  221,  222,  223,  292,  294, 
295,  305,  306,  319,  337,  3;3«, 

341,347,348 

Reduction,  of  fractions     .     .     .     162 

to  common  denominators   .     .     164 

of  surds 263 

Reviews,  23,  42,  65,  83,  107,  121, 
180,  187,  195,  206,  229,  239, 
261,  277,  29(),  307,  313,  318-352 


Roman  niimerals 

Roots  and  coefficients    .    .     . 
Roots,  square      .     .   124,  240,  243. 


5 
282 
247 

Scale  of  signed  numbers   .    .  45 

Servois 15,  84 

Signed  numbers 44 

Signs,  double  use  of 53 

of  aggregation 11 

of  operation 3 

of  qnality 44 

Similar  radicals 268 

teinis 68 

triaTiglcs 193 

Simultaneous  equations    .     .  207 

quadratic  and  linear  ....  297 

three  of  first  degree   ....  224 

two  of  first  degree      ....  208 

two  (juadratic 301 

Solution  of  equations,  1,  25,  27, 
29,  31,  54,  147,  181,  207,  208. 

211,  224,  237.  256.  278,  289 
Solution  of  problems,   hints 

oil 33.37 

Special    products  and    quo- 
tients    108 


INDEX 


373 


[References 

Square,  of  binomial 108 

of  any  polynomial       ....     110 

of  a  trinomial 114 

Square  root 124,  240 

application  of 257 

approximate 251 

of  fractions 25.3 

of  products 241 

of  polynomials 243 

Stevins 161 

Substitution,  elimination  by    .    211 

Subtraction 8,  50 

elimination  by 208 

of  fractions 168 

of  imagiuaries 285 

of  polynomials 72 

of  radicals 268 

of  signed  numbers      ....      50 

Sum 9,45,69 

of  two  cubes 118,  128 

Surds 262 

entire 262 

mixed 262 

order  of 262 

quadratic 262,  274 

Symbols,  of  aggregation  ...      11 
of  operation 3,  7 


are  to  papfes.] 
Systems  of  equations 

207,  224,  297,  302 

Terms,  of  a  fraction       ....  160 

of  a  ratio 188 

similar 68 

transposition  of 31 

Triang-les,  problems  on,  2,  3,  92, 
193,  194,  195,  257,  258,  2.V.), 
260,  261,  293,  294,  306,  307, 

341,  342,  347 

similar 193 

Trinomial  squares 124 


Unknovrn  numbers 


25 


Variables 314 

Variation 315 

Vieta   .     .     .     .41,  101,  206,  239,  313 

Vinculum n 

Volumes,  problems  on,  2,  3,  4, 

36,  153,  293,  325 

Wallls 101 

"Widmann 7 

Written  work,  directions  for   .      30 


Zero,  division  by  . 
multiplication  by 


29,  102,  104 
.     .     .      29 


VB  35V32 


460003 


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